2010
2010
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Paper 1, Section I, D
2010 commentLet be a complex power series. State carefully what it means for the power series to have radius of convergence , with .
Suppose the power series has radius of convergence , with . Show that the sequence is unbounded if .
Find the radius of convergence of .
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Paper 1, Section I, E
2010 commentFind the limit of each of the following sequences; justify your answers.
(i)
(ii)
(iii)
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Paper 1, Section II, E
2010 commentDetermine whether the following series converge or diverge. Any tests that you use should be carefully stated.
(a)
(b)
(c)
(d)
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Paper 1, Section II, F
2010 comment(a) State and prove Taylor's theorem with the remainder in Lagrange's form.
(b) Suppose that is a differentiable function such that and for all . Use the result of (a) to prove that
[No property of the exponential function may be assumed.]
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Paper 1, Section II, D
2010 commentDefine what it means for a bounded function to be Riemann integrable.
Show that a monotonic function is Riemann integrable, where .
Prove that if is a decreasing function with as , then and either both diverge or both converge.
Hence determine, for , when converges.
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Paper 1, Section II, F
2010 comment(a) Let and be a function . Define carefully what it means for to be times differentiable at a point .
Consider the function on the real line, with and
(b) Is differentiable at ?
(c) Show that has points of non-differentiability in any neighbourhood of .
(d) Prove that, in any finite interval , the derivative , at the points where it exists, is bounded: where depends on .
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Paper 2, Section I, A
2010 commentFind the general solutions to the following difference equations for .
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Paper 2, Section I, A
2010 commentLet where the variables and are related by a smooth, invertible transformation. State the chain rule expressing the derivatives and in terms of and and use this to deduce that
where and are second-order partial derivatives, to be determined.
Using the transformation and in the above identity, or otherwise, find the general solution of
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Paper 2, Section II, A
2010 comment(a) Find the general solution of the system of differential equations
(b) Depending on the parameter , find the general solution of the system of differential equations
and explain why has a particular solution of the form with constant vector for but not for .
[Hint: decompose in terms of the eigenbasis of the matrix in (1).]
(c) For , find the solution of (2) which goes through the point at .
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Paper 2, Section II, A
2010 comment(a) Consider the differential equation
with and . Show that is a solution if and only if where
Show further that is also a solution of if is a root of the polynomial of multiplicity at least 2 .
(b) By considering , or otherwise, find the general real solution for satisfying
By using a substitution of the form in , or otherwise, find the general real solution for , with positive, where
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Paper 2, Section II,
2010 comment(a) By using a power series of the form
or otherwise, find the general solution of the differential equation
(b) Define the Wronskian for a second order linear differential equation
and show that . Given a non-trivial solution of show that can be used to find a second solution of and give an expression for in the form of an integral.
(c) Consider the equation (2) with
where and have Taylor expansions
with a positive integer. Find the roots of the indicial equation for (2) with these assumptions. If is a solution, use the method of part (b) to find the first two terms in a power series expansion of a linearly independent solution , expressing the coefficients in terms of and .
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Paper 2, Section II, A
2010 comment(a) State how the nature of a critical (or stationary) point of a function with can be determined by consideration of the eigenvalues of the Hessian matrix of , assuming is non-singular.
(b) Let . Find all the critical points of the function and determine their nature. Determine the zero contour of and sketch a contour plot showing the behaviour of the contours in the neighbourhood of the critical points.
(c) Now let . Show that is a critical point of for which the Hessian matrix of is singular. Find an approximation for to lowest non-trivial order in the neighbourhood of the point . Does have a maximum or a minimum at ? Justify your answer.
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Paper 4, Section I, B
2010 commentA particle of mass and charge moves with trajectory in a constant magnetic field . Write down the Lorentz force on the particle and use Newton's Second Law to deduce that
where is a constant vector and is to be determined. Find and hence for the initial conditions
where and are constants. Sketch the particle's trajectory in the case .
[Unit vectors correspond to a set of Cartesian coordinates. ]
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Paper 4, Section , B
2010 commentLet be an inertial frame with coordinates in two-dimensional spacetime. Write down the Lorentz transformation giving the coordinates in a second inertial frame moving with velocity relative to . If a particle has constant velocity in , find its velocity in . Given that and , show that .
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Paper 4 , Section II, B
2010 commentA sphere of uniform density has mass and radius . Find its moment of inertia about an axis through its centre.
A marble of uniform density is released from rest on a plane inclined at an angle to the horizontal. Let the time taken for the marble to travel a distance down the plane be: (i) if the plane is perfectly smooth; or (ii) if the plane is rough and the marble rolls without slipping.
Explain, with a clear discussion of the forces acting on the marble, whether or not its energy is conserved in each of the cases (i) and (ii). Show that .
Suppose that the original marble is replaced by a new one with the same mass and radius but with a hollow centre, so that its moment of inertia is for some constant . What is the new value for ?
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Paper 4, Section II, B
2010 commentA particle of unit mass moves in a plane with polar coordinates and components of acceleration . The particle experiences a force corresponding to a potential . Show that
are constants of the motion, where
Sketch the graph of in the cases and .
(a) Assuming and , for what range of values of do bounded orbits exist? Find the minimum and maximum distances from the origin, and , on such an orbit and show that
Prove that the minimum and maximum values of the particle's speed, and , obey
(b) Now consider trajectories with and of either sign. Find the distance of closest approach, , in terms of the impact parameter, , and , the limiting value of the speed as . Deduce that if then, to leading order,
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Paper 4, Section II, B
2010 commentConsider a set of particles with position vectors and masses , where . Particle experiences an external force and an internal force from particle , for each . Stating clearly any assumptions you need, show that
where is the total momentum, is the total external force, is the total angular momentum about a fixed point , and is the total external torque about .
Does the result still hold if the fixed point is replaced by the centre of mass of the system? Justify your answer.
Suppose now that the external force on particle is and that all the particles have the same mass . Show that
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Paper 4, Section II, B
2010 commentA particle of rest mass is fired at an identical particle which is stationary in the laboratory. On impact, and annihilate and produce two massless photons whose energies are equal. Assuming conservation of four-momentum, show that the angle between the photon trajectories is given by
where is the relativistic energy of .
Let be the speed of the incident particle . For what value of will the photons move in perpendicular directions? If is very small compared with , show that
[All quantities referred to are measured in the laboratory frame.]
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Paper 3, Section I, D
2010 commentWrite down the matrix representing the following transformations of :
(i) clockwise rotation of around the axis,
(ii) reflection in the plane ,
(iii) the result of first doing (i) and then (ii).
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Paper 3, Section I, D
2010 commentExpress the element in as a product of disjoint cycles. Show that it is in . Write down the elements of its conjugacy class in .
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Paper 3, Section II, D
2010 comment(i) State the orbit-stabilizer theorem.
Let be the group of rotations of the cube, the set of faces. Identify the stabilizer of a face, and hence compute the order of .
Describe the orbits of on the set of pairs of faces.
(ii) Define what it means for a subgroup of to be normal. Show that has a normal subgroup of order 4 .
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Paper 3, Section II, D
2010 commentState Lagrange's theorem. Let be a prime number. Prove that every group of order is cyclic. Prove that every abelian group of order is isomorphic to either or
Show that , the dihedral group of order 12 , is not isomorphic to the alternating .
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Paper 3, Section II, D
2010 commentLet be a group, a set on which acts transitively, the stabilizer of a point .
Show that if stabilizes the point , then there exists an with .
Let , acting on by Möbius transformations. Compute , the stabilizer of . Given
compute the set of fixed points
Show that every element of is conjugate to an element of .
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Paper 3, Section II, D
2010 commentLet be a finite group, the set of proper subgroups of . Show that conjugation defines an action of on .
Let be a proper subgroup of . Show that the orbit of on containing has size at most the index . Show that there exists a which is not conjugate to an element of .
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Paper 4, Section I, E
2010 comment(a) Find the smallest residue which equals .
[You may use any standard theorems provided you state them correctly.]
(b) Find all integers which satisfy the system of congruences
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Paper 4, Section I,
2010 comment(a) Let be a real root of the polynomial , with integer coefficients and leading coefficient 1 . Show that if is rational, then is an integer.
(b) Write down a series for . By considering for every natural number , show that is irrational.
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Paper 4, Section II, E
2010 commentThe Fibonacci numbers are defined for all natural numbers by the rules
Prove by induction on that, for any ,
Deduce that
Put and for . Show that these (Lucas) numbers satisfy
Show also that, for all , the greatest common divisor is 1 , and that the greatest common divisor is at most 2 .
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Paper 4, Section II,
2010 commentState and prove Fermat's Little Theorem.
Let be an odd prime. If , show that divides for infinitely many natural numbers .
Hence show that divides infinitely many of the integers
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Paper 4, Section II,
2010 comment(a) Let be finite non-empty sets, with . Show that there are mappings from to . How many of these are injective ?
(b) State the Inclusion-Exclusion principle.
(c) Prove that the number of surjective mappings from a set of size onto a set of size is
Deduce that
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Paper 4 , Section II, E
2010 commentWhat does it mean for a set to be countable ?
Show that is countable, but is not. Show also that the union of two countable sets is countable.
A subset of has the property that, given and , there exist reals with and with and . Can be countable ? Can be uncountable ? Justify your answers.
A subset of has the property that given there exists such that if for some , then . Is countable ? Justify your answer.
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Paper 2, Section I, F
2010 commentJensen's inequality states that for a convex function and a random variable with a finite mean, .
(a) Suppose that where is a positive integer, and is a random variable taking values with equal probabilities, and where the sum . Deduce from Jensen's inequality that
(b) horses take part in races. The results of different races are independent. The probability for horse to win any given race is , with .
Let be the probability that a single horse wins all races. Express as a polynomial of degree in the variables .
By using (1) or otherwise, prove that .
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Paper 2, Section I, F
2010 commentLet and be two non-constant random variables with finite variances. The correlation coefficient is defined by
(a) Using the Cauchy-Schwarz inequality or otherwise, prove that
(b) What can be said about the relationship between and when either (i) or (ii) . [Proofs are not required.]
(c) Take and let be independent random variables taking values with probabilities . Set
Find .
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Paper 2, Section II, F
2010 comment(a) What does it mean to say that a random variable with values has a geometric distribution with a parameter where ?
An expedition is sent to the Himalayas with the objective of catching a pair of wild yaks for breeding. Assume yaks are loners and roam about the Himalayas at random. The probability that a given trapped yak is male is independent of prior outcomes. Let be the number of yaks that must be caught until a breeding pair is obtained. (b) Find the expected value of . (c) Find the variance of .
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Paper 2, Section II, F
2010 commentThe yearly levels of water in the river Camse are independent random variables , with a given continuous distribution function and . The levels have been observed in years and their values recorded. The local council has decided to construct a dam of height
Let be the subsequent time that elapses before the dam overflows:
(a) Find the distribution function , and show that the mean value
(b) Express the conditional probability , where and , in terms of .
(c) Show that the unconditional probability
(d) Determine the mean value .
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Paper 2, Section II, F
2010 commentIn a branching process every individual has probability of producing exactly offspring, , and the individuals of each generation produce offspring independently of each other and of individuals in preceding generations. Let represent the size of the th generation. Assume that and and let be the generating function of . Thus
(a) Prove that
(b) State a result in terms of about the probability of eventual extinction. [No proofs are required.]
(c) Suppose the probability that an individual leaves descendants in the next generation is , for . Show from the result you state in (b) that extinction is certain. Prove further that in this case
and deduce the probability that the th generation is empty.
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Paper 2, Section II, F
2010 commentLet be bivariate normal random variables, with the joint probability density function
where
and .
(a) Deduce that the marginal probability density function
(b) Write down the moment-generating function of in terms of and proofs are required.]
(c) By considering the ratio prove that, conditional on , the distribution of is normal, with mean and variance and , respectively.
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Paper 3, Section I, C
2010 commentConsider the vector field
defined on all of except the axis. Compute on the region where it is defined.
Let be the closed curve defined by the circle in the -plane with centre and radius 1 , and be the closed curve defined by the circle in the -plane with centre and radius 1 .
By using your earlier result, or otherwise, evaluate the line integral .
By explicit computation, evaluate the line integral . Is your result consistent with Stokes' theorem? Explain your answer briefly.
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Paper 3, Section I, C
2010 commentA curve in two dimensions is defined by the parameterised Cartesian coordinates
where the constants . Sketch the curve segment corresponding to the range . What is the length of the curve segment between the points and , as a function of ?
A geometrically sensitive ant walks along the curve with varying speed , where is the curvature at the point corresponding to parameter . Find the time taken by the ant to walk from to , where is a positive integer, and hence verify that this time is independent of .
[You may quote without proof the formula ]
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Paper 3, Section II, C
2010 comment(a) Define a rank two tensor and show that if two rank two tensors and are the same in one Cartesian coordinate system, then they are the same in all Cartesian coordinate systems.
The quantity has the property that, for every rank two tensor , the quantity is a scalar. Is necessarily a rank two tensor? Justify your answer with a proof from first principles, or give a counterexample.
(b) Show that, if a tensor is invariant under rotations about the -axis, then it has the form
(c) The inertia tensor about the origin of a rigid body occupying volume and with variable mass density is defined to be
The rigid body has uniform density and occupies the cylinder
Show that the inertia tensor of about the origin is diagonal in the coordinate system, and calculate its diagonal elements.
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Paper 3, Section II, C
2010 commentLet be a function of two variables, and a region in the -plane. State the rule for evaluating as an integral with respect to new variables and .
Sketch the region in the -plane defined by
Sketch the corresponding region in the -plane, where
Express the integral
as an integral with respect to and . Hence, or otherwise, calculate .
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Paper 3 , Section II, C
2010 commentState the divergence theorem (also known as Gauss' theorem) relating the surface and volume integrals of appropriate fields.
The surface is defined by the equation for ; the surface is defined by the equation for ; the surface is defined by the equation for satisfying . The surface is defined to be the union of the surfaces and . Sketch the surfaces and (hence) .
The vector field is defined by
Evaluate the integral
where the surface element points in the direction of the outward normal to .
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Paper 3, Section II, C
2010 commentGiven a spherically symmetric mass distribution with density , explain how to obtain the gravitational field , where the potential satisfies Poisson's equation
The remarkable planet Geometria has radius 1 and is composed of an infinite number of stratified spherical shells labelled by integers . The shell has uniform density , where is a constant, and occupies the volume between radius and .
Obtain a closed form expression for the mass of Geometria.
Obtain a closed form expression for the gravitational field due to Geometria at a distance from its centre of mass, for each positive integer . What is the potential due to Geometria for ?
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Paper 1, Section I,
2010 commentLet be the matrix representing a linear map with respect to the bases of and of , so that . Let be another basis of and let be another basis of . Show that the matrix representing with respect to these new bases satisfies with matrices and which should be defined.
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Paper 1, Section I, C
2010 comment(a) The complex numbers and satisfy the equations
What are the possible values of ? Justify your answer.
(b) Show that for all complex numbers and . Does the inequality hold for all complex numbers and ? Justify your answer with a proof or a counterexample.
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Paper 1, Section II, A
2010 commentLet and be real matrices.
(i) Define the trace of , and show that .
(ii) Show that , with if and only if is the zero matrix. Hence show that
Under what condition on and is equality achieved?
(iii) Find a basis for the subspace of matrices such that
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Paper 1, Section II,
2010 commentLet and be vectors in . Give a definition of the dot product, , the cross product, , and the triple product, . Explain what it means to say that the three vectors are linearly independent.
Let and be vectors in . Let be a matrix with entries . Show that
Hence show that is of maximal rank if and only if the sets of vectors , and are both linearly independent.
Now let and be sets of vectors in , and let be an matrix with entries . Is it the case that is of maximal rank if and only if the sets of vectors and are both linearly independent? Justify your answer with a proof or a counterexample.
Given an integer , is it always possible to find a set of vectors in with the property that every pair is linearly independent and that every triple is linearly dependent? Justify your answer.
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Paper 1, Section II, B
2010 commentLet be a complex matrix with an eigenvalue . Show directly from the definitions that:
(i) has an eigenvalue for any integer ; and
(ii) if is invertible then and has an eigenvalue .
For any complex matrix , let . Using standard properties of determinants, show that:
(iii) ; and
(iv) if is invertible,
Explain, including justifications, the relationship between the eigenvalues of and the polynomial .
If has an eigenvalue , does it follow that has an eigenvalue with ? Give a proof or counterexample.
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Paper 1, Section II, B
2010 commentLet be a real orthogonal matrix with a real eigenvalue corresponding to some real eigenvector. Show algebraically that and interpret this result geometrically.
Each of the matrices
has an eigenvalue . Confirm this by finding as many independent eigenvectors as possible with this eigenvalue, for each matrix in turn.
Show that one of the matrices above represents a rotation, and find the axis and angle of rotation. Which of the other matrices represents a reflection, and why?
State, with brief explanations, whether the matrices are diagonalisable (i) over the real numbers; (ii) over the complex numbers.
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Paper 2, Section I, G
2010 commentLet be a real number, and let be the space of sequences of real numbers with convergent. Show that defines a norm on .
Let denote the space of sequences a with bounded; show that . If , show that the norms on given by restricting to the norms on and on are not Lipschitz equivalent.
By considering sequences of the form in , for an appropriate real number, or otherwise, show that (equipped with the norm ) is not complete.
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Paper 3, Section I, G
2010 commentConsider the map given by
Show that is differentiable everywhere and find its derivative.
Stating carefully any theorem that you quote, show that is locally invertible near a point unless .
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Paper 4, Section I, G
2010 commentLet denote the set of continuous real-valued functions on the interval . For , set
Show that both and define metrics on . Does the identity map on define a continuous map of metric spaces Does the identity map define a continuous map of metric spaces ?
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Paper 1, Section II, G
2010 commentState and prove the contraction mapping theorem. Demonstrate its use by showing that the differential equation , with boundary condition , has a unique solution on , with one-sided derivative at zero.
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Paper 2, Section II, G
2010 commentSuppose the functions are defined on the open interval and that tends uniformly on to a function . If the are continuous, show that is continuous. If the are differentiable, show by example that need not be differentiable.
Assume now that each is differentiable and the derivatives converge uniformly on . For any given , we define functions by
Show that each is continuous. Using the general principle of uniform convergence (the Cauchy criterion) and the Mean Value Theorem, or otherwise, prove that the functions converge uniformly to a continuous function on , where
Deduce that is differentiable on .
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Paper 3, Section II, G
2010 commentLet be a map on an open subset . Explain what it means for to be differentiable on . If is a differentiable map on an open subset with , state and prove the Chain Rule for the derivative of the composite .
Suppose now is a differentiable function for which the partial derivatives for all . By considering the function given by
or otherwise, show that there exists a differentiable function with at all points of
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Paper 4, Section II, G
2010 commentWhat does it mean to say that a function on an interval in is uniformly continuous? Assuming the Bolzano-Weierstrass theorem, show that any continuous function on a finite closed interval is uniformly continuous.
Suppose that is a continuous function on the real line, and that tends to finite limits as ; show that is uniformly continuous.
If is a uniformly continuous function on , show that is bounded as . If is a continuous function on for which as , determine whether is necessarily uniformly continuous, giving proof or counterexample as appropriate.
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Paper 4, Section I, G
2010 commentState the principle of the argument for meromorphic functions and show how it follows from the Residue theorem.
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Paper 3, Section II, G
2010 commentState Morera's theorem. Suppose are analytic functions on a domain and that tends locally uniformly to on . Show that is analytic on . Explain briefly why the derivatives tend locally uniformly to .
Suppose now that the are nowhere vanishing and is not identically zero. Let be any point of ; show that there exists a closed disc with centre , on which the convergence of and are both uniform, and where is nowhere zero on . By considering
(where denotes the boundary of ), or otherwise, deduce that .
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Paper 1, Section II, A
2010 commentCalculate the following real integrals by using contour integration. Justify your steps carefully.
(a)
(b)
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Paper 2, Section II, A
2010 comment(a) Prove that a complex differentiable map, , is conformal, i.e. preserves angles, provided a certain condition holds on the first complex derivative of .
(b) Let be the region
Draw the region . It might help to consider the two sets
(c) For the transformations below identify the images of .
Step 1: The first map is ,
Step 2: The second map is the composite where ,
Step 3: The third map is the composite where .
(d) Write down the inverse map to the composite , explaining any choices of branch.
[The composite means .]
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Paper 3, Section I, A
2010 comment(a) Prove that the real and imaginary parts of a complex differentiable function are harmonic.
(b) Find the most general harmonic polynomial of the form
where and are real.
(c) Write down a complex analytic function of of which is the real part.
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Paper 1, Section I, A
2010 comment(a) Write down the definition of the complex derivative of the function of a single complex variable.
(b) Derive the Cauchy-Riemann equations for the real and imaginary parts and of , where and
(c) State necessary and sufficient conditions on and for the function to be complex differentiable.
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Paper 4, Section II, A
2010 commentA linear system is described by the differential equation
with initial conditions
The Laplace transform of is defined as
You may assume the following Laplace transforms,
(a) Use Laplace transforms to determine the response, , of the system to the signal
(b) Determine the response, , given that its Laplace transform is
(c) Given that
leads to the response with Laplace transform
determine .
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Paper 2, Section I,
2010 commentWrite down Maxwell's equations for electromagnetic fields in a non-polarisable and non-magnetisable medium.
Show that the homogenous equations (those not involving charge or current densities) can be solved in terms of vector and scalar potentials and .
Then re-express the inhomogeneous equations in terms of and . Show that the potentials can be chosen so as to set and hence rewrite the inhomogeneous equations as wave equations for the potentials. [You may assume that the inhomogeneous wave equation always has a solution for any given .]
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Paper 4, Section I, B
2010 commentGive an expression for the force on a charge moving at velocity in electric and magnetic fields and . Consider a stationary electric circuit , and let be a stationary surface bounded by . Derive from Maxwell's equations the result
where the electromotive force and the flux .
Now assume that also holds for a moving circuit. A circular loop of wire of radius and total resistance , whose normal is in the -direction, moves at constant speed in the -direction in the presence of a magnetic field . Find the current in the wire.
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Paper 1, Section II, C
2010 commentA capacitor consists of three perfectly conducting coaxial cylinders of radii and where , and length where so that end effects may be ignored. The inner and outer cylinders are maintained at zero potential, while the middle cylinder is held at potential . Assuming its cylindrical symmetry, compute the electrostatic potential within the capacitor, the charge per unit length on the middle cylinder, the capacitance and the electrostatic energy, both per unit length.
Next assume that the radii and are fixed, as is the potential , while the radius is allowed to vary. Show that the energy achieves a minimum when is the geometric mean of and .
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Paper 2, Section II, C
2010 commentA steady current flows around a loop of a perfectly conducting narrow wire. Assuming that the gauge condition holds, the vector potential at points away from the loop may be taken to be
First verify that the gauge condition is satisfied here. Then obtain the Biot-Savart formula for the magnetic field
Next suppose there is a similar but separate loop with current . Show that the magnetic force exerted on loop by loop is
Is this consistent with Newton's third law? Justify your answer.
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Paper 3, Section II, C
2010 commentWrite down Maxwell's equations in a region with no charges and no currents. Show that if and is a solution then so is and . Write down the boundary conditions on and at the boundary with unit normal between a perfect conductor and a vacuum.
Electromagnetic waves propagate inside a tube of perfectly conducting material. The tube's axis is in the -direction, and it is surrounded by a vacuum. The fields may be taken to be the real parts of
Write down Maxwell's equations in terms of and .
Suppose first that . Show that the solution is determined by
where the function satisfies
and vanishes on the boundary of the tube. Here is a constant whose value should be determined.
Obtain a similar condition for the case where . [You may find it useful to use a result from the first paragraph.] What is the corresponding boundary condition?
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Paper 1, Section I, B
2010 commentA planar solenoidal velocity field has the velocity potential
Find and sketch (i) the streamlines at ; (ii) the pathline that passes through the origin at ; (iii) the locus at of points that pass through the origin at earlier times (streakline).
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Paper 2, Section I, B
2010 commentWrite down an expression for the velocity field of a line vortex of strength .
Consider identical line vortices of strength arranged at equal intervals round a circle of radius . Show that the vortices all move around the circle at constant angular velocity .
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Paper 1, Section II, B
2010 commentStarting with the Euler equations for an inviscid incompressible fluid, derive Bernoulli's theorem for unsteady irrotational flow.
Inviscid fluid of density is contained within a U-shaped tube with the arms vertical, of height and with the same (unit) cross-section. The ends of the tube are closed. In the equilibrium state the pressures in the two arms are and and the heights of the fluid columns are .
The fluid in arm 1 is displaced upwards by a distance (and in the other arm downward by the same amount). In the subsequent evolution the pressure above each column may be taken as inversely proportional to the length of tube above the fluid surface. Using Bernoulli's theorem, show that obeys the equation
Now consider the special case . Construct a first integral of this equation and hence give an expression for the total kinetic energy of the flow in terms of and the maximum displacement .
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Paper 3, Section II, B
2010 commentWrite down the exact kinematic and dynamic boundary conditions that apply at the free surface of a fluid layer in the presence of gravity in the -direction. Show how these may be approximated for small disturbances of a hydrostatic state about . (The flow of the fluid is in the -plane and may be taken to be irrotational, and the pressure at the free surface may be assumed to be constant.)
Fluid of density fills the region . At the -component of the velocity is , where . Find the resulting disturbance of the free surface, assuming this to be small. Explain physically why your answer has a singularity for a particular value of .
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Paper 4, Section II, B
2010 commentWrite down the velocity potential for a line source flow of strength located at in polar coordinates and derive the velocity components .
A two-dimensional flow field consists of such a source in the presence of a circular cylinder of radius centred at the origin. Show that the flow field outside the cylinder is the sum of the original source flow, together with that due to a source of the same strength at and another at the origin, of a strength to be determined.
Use Bernoulli's law to find the pressure distribution on the surface of the cylinder, and show that the total force exerted on it is in the -direction and of magnitude
where is the density of the fluid. Without evaluating the integral, show that it is positive. Comment on the fact that the force on the cylinder is therefore towards the source.
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Paper 1, Section I, F
2010 comment(i) Define the notion of curvature for surfaces embedded in .
(ii) Prove that the unit sphere in has curvature at all points.
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Paper 3, Section I, F
2010 comment(i) Write down the Poincaré metric on the unit disc model of the hyperbolic plane. Compute the hyperbolic distance from to , with .
(ii) Given a point in and a hyperbolic line in with not on , describe how the minimum distance from to is calculated. Justify your answer.
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Paper 2, Section II, F
2010 commentSuppose that and that is the half-cone defined by , . By using an explicit smooth parametrization of , calculate the curvature of .
Describe the geodesics on . Show that for , no geodesic intersects itself, while for some geodesic does so.
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Paper 3, Section II, F
2010 commentDescribe the hyperbolic metric on the upper half-plane . Show that any Möbius transformation that preserves is an isometry of this metric.
Suppose that are distinct and that the hyperbolic line through and meets the real axis at . Show that the hyperbolic distance between and is given by , where is the cross-ratio of the four points , taken in an appropriate order.
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Paper 4, Section II, F
2010 commentSuppose that is the unit disc, with Riemannian metric
[Note that this is not a multiple of the Poincaré metric.] Show that the diameters of are, with appropriate parametrization, geodesics.
Show that distances between points in are bounded, but areas of regions in are unbounded.
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Paper 2, Section I,
2010 commentGive the definition of conjugacy classes in a group . How many conjugacy classes are there in the symmetric group on four letters? Briefly justify your answer.
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Paper 3, Section I, H
2010 commentLet be the ring of integers or the polynomial ring . In each case, give an example of an ideal of such that the quotient ring has a non-trivial idempotent (an element with and ) and a non-trivial nilpotent element (an element with and for some positive integer ). Exhibit these elements and justify your answer.
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Paper 4, Section I, H
2010 commentLet be a free -module generated by and . Let be two non-zero integers, and be the submodule of generated by . Prove that the quotient module is free if and only if are coprime.
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Paper 1, Section II, H
2010 commentProve that the kernel of a group homomorphism is a normal subgroup of the group .
Show that the dihedral group of order 8 has a non-normal subgroup of order 2. Conclude that, for a group , a normal subgroup of a normal subgroup of is not necessarily a normal subgroup of .
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Paper 2, Section II, H
2010 commentFor ideals of a ring , their product is defined as the ideal of generated by the elements of the form where and .
(1) Prove that, if a prime ideal of contains , then contains either or .
(2) Give an example of and such that the two ideals and are different from each other.
(3) Prove that there is a natural bijection between the prime ideals of and the prime ideals of .
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Paper 3, Section II, H
2010 commentLet be an integral domain and its group of units. An element of is irreducible if it is not a product of two elements in . When is Noetherian, show that every element of is a product of finitely many irreducible elements of .
-
Paper 4, Section II,
2010 commentLet , a 2-dimensional vector space over the field , and let
(1) List all 1-dimensional subspaces of in terms of . (For example, there is a subspace generated by
(2) Consider the action of the matrix group
on the finite set of all 1-dimensional subspaces of . Describe the stabiliser group of . What is the order of ? What is the order of ?
(3) Let be the subgroup of all elements of which act trivially on . Describe , and prove that is isomorphic to , the symmetric group on four letters.
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Paper 1, Section I, F
2010 commentSuppose that is the complex vector space of polynomials of degree at most in the variable . Find the Jordan normal form for each of the linear transformations and acting on .
-
Paper 2, Section I, F
2010 commentSuppose that is an endomorphism of a finite-dimensional complex vector space.
(i) Show that if is an eigenvalue of , then is an eigenvalue of .
(ii) Show conversely that if is an eigenvalue of , then there is an eigenvalue of with .
-
Paper 4, Section I, F
2010 commentDefine the notion of an inner product on a finite-dimensional real vector space , and the notion of a self-adjoint linear map .
Suppose that is the space of real polynomials of degree at most in a variable . Show that
is an inner product on , and that the map :
is self-adjoint.
-
Paper 1, Section II, F
2010 commentLet denote the vector space of real matrices.
(1) Show that if , then is a positive-definite symmetric bilinear form on .
(2) Show that if , then is a quadratic form on . Find its rank and signature.
[Hint: Consider symmetric and skew-symmetric matrices.]
-
Paper 2, Section II, F
2010 comment(i) Show that two complex matrices are similar (i.e. there exists invertible with ) if and only if they represent the same linear map with respect to different bases.
(ii) Explain the notion of Jordan normal form of a square complex matrix.
(iii) Show that any square complex matrix is similar to its transpose.
(iv) If is invertible, describe the Jordan normal form of in terms of that of .
Justify your answers.
-
Paper 3, Section II, F
2010 commentSuppose that is a finite-dimensional vector space over , and that is a -linear map such that for some . Show that if is a subspace of such that , then there is a subspace of such that and .
[Hint: Show, for example by picking bases, that there is a linear map with for all . Then consider with
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Paper 4, Section II, F
2010 comment(i) Show that the group of orthogonal real matrices has a normal subgroup .
(ii) Show that if and only if is odd.
(iii) Show that if is even, then is not the direct product of with any normal subgroup.
[You may assume that the only elements of that commute with all elements of are .]
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Paper 3, Section I, E
2010 commentAn intrepid tourist tries to ascend Springfield's famous infinite staircase on an icy day. When he takes a step with his right foot, he reaches the next stair with probability , otherwise he falls down and instantly slides back to the bottom with probability . Similarly, when he steps with his left foot, he reaches the next stair with probability , or slides to the bottom with probability . Assume that he always steps first with his right foot when he is at the bottom, and alternates feet as he ascends. Let be his position after his th step, so that when he is on the stair , where 0 is the bottom stair.
(a) Specify the transition probabilities for the Markov chain for any .
(b) Find the equilibrium probabilities , for . [Hint:
(c) Argue that the chain is irreducible and aperiodic and evaluate the limit
for each .
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Paper 4, Section I, E
2010 commentConsider a Markov chain with state space and transition probabilities given by the following table.
\begin{tabular}{c|cccc} & & & & \ \hline & & & & 0 \ & 0 & & 0 & \ & & 0 & & \ & 0 & & 0 & \end{tabular}
By drawing an appropriate diagram, determine the communicating classes of the chain, and classify them as either open or closed. Compute the following transition and hitting probabilities:
-
for a fixed
-
for some .
-
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Paper 1, Section II, E
2010 commentLet be a Markov chain.
(a) What does it mean to say that a state is positive recurrent? How is this property related to the equilibrium probability ? You do not need to give a full proof, but you should carefully state any theorems you use.
(b) What is a communicating class? Prove that if states and are in the same communicating class and is positive recurrent then is positive recurrent also.
A frog is in a pond with an infinite number of lily pads, numbered She hops from pad to pad in the following manner: if she happens to be on pad at a given time, she hops to one of pads with equal probability.
(c) Find the equilibrium distribution of the corresponding Markov chain.
(d) Now suppose the frog starts on pad and stops when she returns to it. Show that the expected number of times the frog hops is ! where What is the expected number of times she will visit the lily pad ?
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Paper 2, Section II, E
2010 commentLet be a simple, symmetric random walk on the integers , with and . For each integer , let . Show that is a stopping time.
Define a random variable by the rule
Show that is also a simple, symmetric random walk.
Let . Explain why for . By using the process constructed above, show that, for ,
and thus
Hence compute
when and are positive integers with . [Hint: if is even, then must be even, and if is odd, then must be odd.]
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Paper 2, Section I, A
2010 commentConsider the initial value problem
where is a second-order linear operator involving differentiation with respect to . Explain briefly how to solve this by using a Green's function.
Now consider
where is a constant, subject to the same initial conditions. Solve this using the Green's function, and explain how your answer is related to a problem in Newtonian dynamics.
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Paper 3, Section I, B
2010 commentShow that Laplace's equation in polar coordinates has solutions proportional to for any constant .
Find the function satisfying Laplace's equation in the region , where .
[The Laplacian in polar coordinates is
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Paper 4, Section I, A
2010 comment(a) By considering strictly monotonic differentiable functions , such that the zeros satisfy but , establish the formula
Hence show that for a general differentiable function with only such zeros, labelled by ,
(b) Hence by changing to plane polar coordinates, or otherwise, evaluate,
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Paper 1, Section II, A
2010 comment(a) A function is periodic with period and has continuous derivatives up to and including the th derivative. Show by integrating by parts that the Fourier coefficients of
decay at least as fast as as
(b) Calculate the Fourier series of on .
(c) Comment on the decay rate of your Fourier series.
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Paper 2, Section II, B
2010 commentExplain briefly the use of the method of characteristics to solve linear first-order partial differential equations.
Use the method to solve the problem
where is a constant, with initial condition .
By considering your solution explain:
(i) why initial conditions cannot be specified on the whole -axis;
(ii) why a single-valued solution in the entire plane is not possible if .
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Paper 3, Section II, A
2010 comment(a) Put the equation
into Sturm-Liouville form.
(b) Suppose are eigenfunctions such that are bounded as tends to zero and
Identify the weight function and the most general boundary conditions on which give the orthogonality relation
(c) The equation
has a solution and a second solution which is not bounded at the origin. The zeros of arranged in ascending order are . Given that , show that the eigenvalues of the Sturm-Liouville problem in (b) are
(d) Using the differential equations for and and integration by parts, show that
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Paper 4, Section II, B
2010 commentDefining the function , prove Green's third identity for functions satisfying Laplace's equation in a volume with surface , namely
A solution is sought to the Neumann problem for in the half plane :
where . It is assumed that . Explain why this condition is necessary.
Construct an appropriate Green's function satisfying at , using the method of images or otherwise. Hence find the solution in the form
where is to be determined.
Now let
By expanding in inverse powers of , show that
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Paper 2, Section I, H
2010 commentOn the set of rational numbers, the 3 -adic metric is defined as follows: for , define and , where is the integer satisfying where is a rational number whose denominator and numerator are both prime to 3 .
(1) Show that this is indeed a metric on .
(2) Show that in , we have as while as . Let be the usual metric on . Show that neither the identity map nor its inverse is continuous.
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Paper 3, Section I, H
2010 commentLet be a topological space and be a set. Let be a surjection. The quotient topology on is defined as follows: a subset is open if and only if is open in .
(1) Show that this does indeed define a topology on , and show that is continuous when we endow with this topology.
(2) Let be another topological space and be a map. Show that is continuous if and only if is continuous.
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Paper 1, Section II, H
2010 commentLet and be continuous maps of topological spaces with .
(1) Suppose that (i) is path-connected, and (ii) for every , its inverse image is path-connected. Prove that is path-connected.
(2) Prove the same statement when "path-connected" is everywhere replaced by "connected".
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Paper 4, Section II, H
2010 comment(1) Prove that if is a compact topological space, then a closed subset of endowed with the subspace topology is compact.
(2) Consider the following equivalence relation on :
Let be the quotient space endowed with the quotient topology, and let be the canonical surjection mapping each element to its equivalence class. Let
(i) Show that is compact.
(ii) Assuming that is dense in , show that is bijective but not homeomorphic.
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Paper 1, Section I, C
2010 commentObtain the Cholesky decompositions of
What is the minimum value of for to be positive definite? Verify that if then is positive definite.
-
Paper 4, Section I, C
2010 commentSuppose are pointwise distinct and is continuous on . For define
and for
Show that is a polynomial of order which interpolates at .
Given and , determine the interpolating polynomial.
-
Paper 1, Section II, 18C
2010 commentLet
be an inner product. The Hermite polynomials are polynomials in of degree with leading term which are orthogonal with respect to the inner product, with
and . Find a three-term recurrence relation which is satisfied by and for . [You may assume without proof that
Next let be the distinct zeros of and for define the Lagrangian polynomials
associated with these points. Prove that if .
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Paper 2, Section II, C
2010 commentConsider the initial value problem for an autonomous differential equation
and its approximation on a grid of points . Writing , it is proposed to use one of two Runge-Kutta schemes defined by
where and
What is the order of each scheme? Determine the -stability of each scheme.
-
Paper 3, Section II, C
2010 commentDefine the QR factorization of an matrix and explain how it can be used to solve the least squares problem of finding the which minimises where , and the norm is the Euclidean one.
Define a Householder (reflection) transformation and show that it is an orthogonal matrix.
Using a Householder reflection, solve the least squares problem for
giving both and .
-
Paper 1, Section I, 8E
2010 commentWhat is the maximal flow problem in a network?
Explain the Ford-Fulkerson algorithm. Why must this algorithm terminate if the initial flow is set to zero and all arc capacities are rational numbers?
-
Paper 2, Section I, E
2010 commentConsider the function defined by
Use the Lagrangian sufficiency theorem to evaluate . Compute the derivative .
-
Paper 3 , Section II, E
2010 commentLet be the payoff matrix of a two-person, zero-sum game. What is Player I's optimization problem?
Write down a sufficient condition that a vector is an optimal mixed strategy for Player I in terms of the optimal mixed strategy of Player II and the value of the game. If and is an invertible, symmetric matrix such that , where , show that the value of the game is
Consider the following game: Players I and II each have three cards labelled 1,2 , and 3. Each player chooses one of her cards, independently of the other, and places it in the same envelope. If the sum of the numbers in the envelope is smaller than or equal to 4, then Player II pays Player I the sum (in ), and otherwise Player I pays Player II the sum. (For instance, if Player I chooses card 3 and Player II choose card 2, then Player I pays Player II £5.) What is the optimal strategy for each player?
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Paper 4, Section II, E
2010 commentA factory produces three types of sugar, types , , and , from three types of syrup, labelled , and C. The following table contains the number of litres of syrup necessary to make each kilogram of sugar.
\begin{tabular}{c|ccc} & & & \ \hline & 3 & 2 & 1 \ & 2 & 3 & 2 \ & 4 & 1 & 2 \end{tabular}
For instance, one kilogram of type sugar requires 3 litres of litres of , and 4 litres of C. The factory can sell each type of sugar for one pound per kilogram. Assume that the factory owner can use no more than 44 litres of and 51 litres of , but is required by law to use at least 12 litres of C. If her goal is to maximize profit, how many kilograms of each type of sugar should the factory produce?
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Paper 3, Section I, D
2010 commentWrite down the commutation relations between the components of position and momentum for a particle in three dimensions.
A particle of mass executes simple harmonic motion with Hamiltonian
and the orbital angular momentum operator is defined by
Show that the components of are observables commuting with . Explain briefly why the components of are not simultaneous observables. What are the implications for the labelling of states of the three-dimensional harmonic oscillator?
-
Paper 4, Section I, D
2010 commentDetermine the possible values of the energy of a particle free to move inside a cube of side , confined there by a potential which is infinite outside and zero inside.
What is the degeneracy of the lowest-but-one energy level?
-
Paper 1, Section II, 15D
2010 commentA particle of unit mass moves in one dimension in a potential
Show that the stationary solutions can be written in the form
You should give the value of and derive any restrictions on . Hence determine the possible energy eigenvalues .
The particle has a wave function which is even in at . Write down the general form for , using the fact that is an even function of only if is even. Hence write down and show that its probability density is periodic in time with period .
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Paper 2, Section II, D
2010 commentA particle of mass moves in a one-dimensional potential defined by
where and are positive constants. Defining and , show that for any allowed positive value of the energy with then
Find the minimum value of for this equation to have a solution.
Find the normalized wave function for the particle. Write down an expression for the expectation value of in terms of two integrals, which you need not evaluate. Given that
discuss briefly the possibility of being greater than . [Hint: consider the graph of - ka cot against
-
Paper 3, Section II, D
2010 commentA (a particle of the same charge as the electron but 270 times more massive) is bound in the Coulomb potential of a proton. Assuming that the wave function has the form , where and are constants, determine the normalized wave function of the lowest energy state of the , assuming it to be an -wave (i.e. the state with ). (You should treat the proton as fixed in space.)
Calculate the probability of finding the inside a sphere of radius in terms of the ratio , and show that this probability is given by if is very small. Would the result be larger or smaller if the were in a -wave state? Justify your answer very briefly.
[Hint: in spherical polar coordinates,
-
Paper 1, Section I, E
2010 commentSuppose are independent random variables, where is an unknown parameter. Explain carefully how to construct the uniformly most powerful test of size for the hypothesis versus the alternative .
-
Paper 2, Section I, E
2010 commentA washing powder manufacturer wants to determine the effectiveness of a television advertisement. Before the advertisement is shown, a pollster asks 100 randomly chosen people which of the three most popular washing powders, labelled and , they prefer. After the advertisement is shown, another 100 randomly chosen people (not the same as before) are asked the same question. The results are summarized below.
\begin{tabular}{c|ccc} & & & \ \hline before & 36 & 47 & 17 \ after & 44 & 33 & 23 \end{tabular}
Derive and carry out an appropriate test at the significance level of the hypothesis that the advertisement has had no effect on people's preferences.
[You may find the following table helpful:
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Paper 1, Section II, E
2010 commentConsider the the linear regression model
where the numbers are known, the independent random variables have the distribution, and the parameters and are unknown. Find the maximum likelihood estimator for .
State and prove the Gauss-Markov theorem in the context of this model.
Write down the distribution of an arbitrary linear estimator for . Hence show that there exists a linear, unbiased estimator for such that
for all linear, unbiased estimators .
[Hint: If then
-
Paper 3, Section II, E
2010 commentLet be independent random variables with unknown parameter . Find the maximum likelihood estimator of , and state the distribution of . Show that has the distribution. Find the confidence interval for of the form for a constant depending on .
Now, taking a Bayesian point of view, suppose your prior distribution for the parameter is . Show that your Bayesian point estimator of for the loss function is given by
Find a constant depending on such that the posterior probability that is equal to .
[The density of the distribution is , for
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Paper 4, Section II, E
2010 commentConsider a collection of independent random variables with common density function depending on a real parameter . What does it mean to say is a sufficient statistic for ? Prove that if the joint density of satisfies the factorisation criterion for a statistic , then is sufficient for .
Let each be uniformly distributed on . Find a two-dimensional sufficient statistic . Using the fact that is an unbiased estimator of , or otherwise, find an unbiased estimator of which is a function of and has smaller variance than . Clearly state any results you use.
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Paper 1, Section I, D
2010 comment(a) Define what it means for a function to be convex and strictly convex.
(b) State a necessary and sufficient first-order condition for strict convexity of , and give, with proof, an example of a function which is strictly convex but with second derivative which is not everywhere strictly positive.
-
Paper 1, Section II, G
2010 comment(i) Let . Show that is birational to , but not isomorphic to it.
(ii) Let be an affine variety. Define the dimension of in terms of the tangent spaces of .
(iii) Let be an irreducible polynomial, where is an algebraically closed field of arbitrary characteristic. Show that .
[You may assume the Nullstellensatz.]
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Paper 2, Section II, G
2010 commentLet be the set of matrices of rank at most over a field . Show that is naturally an affine subvariety of and that is a Zariski closed subvariety of .
Show that if , then 0 is a singular point of .
Determine the dimension of .
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Paper 3, Section II, G
2010 comment(i) Let be a curve, and be a smooth point on . Define what a local parameter at is.
Now let be a rational map to a quasi-projective variety . Show that if is projective, extends to a morphism defined at .
Give an example where this fails if is not projective, and an example of a morphism which does not extend to
(ii) Let and be curves in over a field of characteristic not equal to 2 . Let be the map . Determine the degree of , and the ramification for all .
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Paper 4, Section II, G
2010 commentLet be the projective curve obtained from the affine curve , where the are distinct and .
(i) Show there is a unique point at infinity, .
(ii) Compute .
(iii) Show .
(iv) Compute for all .
[You may not use the Riemann-Roch theorem.]
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Paper 1, Section II, H
2010 commentState the path lifting and homotopy lifting lemmas for covering maps. Suppose that is path connected and locally path connected, that and are covering maps, and that and are simply connected. Using the lemmas you have stated, but without assuming the correspondence between covering spaces and subgroups of , prove that is homeomorphic to .
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Paper 2, Section II,
2010 commentLet be the finitely presented group . Construct a path connected space with . Show that has a unique connected double cover , and give a presentation for .
-
Paper 3, Section II, H
2010 commentSuppose is a finite simplicial complex and that is a free abelian group for each value of . Using the Mayer-Vietoris sequence or otherwise, compute in terms of . Use your result to compute .
[Note that , where there are factors in the product.]
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Paper 3, Section II, B
2010 commentState Bloch's theorem for a one dimensional lattice which is invariant under translations by .
A simple model of a crystal consists of a one-dimensional linear array of identical sites with separation . At the th site the Hamiltonian, neglecting all other sites, is and an electron may occupy either of two states, and , where
and and are orthonormal. How are and related to and ?
The full Hamiltonian is and is invariant under translations by . Write trial wavefunctions for the eigenstates of this model appropriate to a tight binding approximation if the electron has probability amplitudes and to be in the states and respectively.
Assume that the only non-zero matrix elements in this model are, for all ,
where and . Show that the time-dependent Schrödinger equation governing the amplitudes becomes
By examining solutions of the form
show that the allowed energies of the electron are two bands given by
Define the Brillouin zone for this system and find the energies at the top and bottom of both bands. Hence, show that the energy gap between the bands is
Show that the wavefunctions satisfy Bloch's theorem.
Describe briefly what are the crucial differences between insulators, conductors and semiconductors.
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Paper 4, Section II,
2010 commentState the Snake Lemma. Explain how to define the boundary map which appears in it, and check that it is well-defined. Derive the Mayer-Vietoris sequence from the Snake Lemma.
Given a chain complex , let be the span of all elements in with grading greater than or equal to , and let be the span of all elements in with grading less than . Give a short exact sequence of chain complexes relating , and . What is the boundary map in the corresponding long exact sequence?
-
Paper 1, Section II, B
2010 commentGive an account of the variational principle for establishing an upper bound on the ground-state energy, , of a particle moving in a potential in one dimension.
Explain how an upper bound on the energy of the first excited state can be found in the case that is a symmetric function.
A particle of mass moves in the potential
Use the trial wavefunction
where is a positive real parameter, to establish the upper bound for the energy of the ground state, where
Show that, for has one zero and find its position.
Show that the turning points of are given by
and deduce that there is one turning point in for all .
Sketch for and hence deduce that has at least one bound state for all .
For show that
where .
[You may use the result that for ]
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Paper 2, Section II, B
2010 commentA beam of particles of mass and momentum is incident along the -axis. Write down the asymptotic form of the wave function which describes scattering under the influence of a spherically symmetric potential and which defines the scattering amplitude .
Given that, for large ,
show how to derive the partial-wave expansion of the scattering amplitude in the form
Obtain an expression for the total cross-section, .
Let have the form
where
Show that the phase-shift satisfies
where .
Assume to be large compared with so that may be approximated by . Show, using graphical methods or otherwise, that there are values for for which for some integer , which should not be calculated. Show that the smallest value, , of for which this condition holds certainly satisfies .
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Paper 4, Section II, B
2010 commentThe scattering amplitude for electrons of momentum incident on an atom located at the origin is where . Explain why, if the atom is displaced by a position vector a, the asymptotic form of the scattering wave function becomes
where and . For electrons incident on atoms in a regular Bravais crystal lattice show that the differential cross-section for scattering in the direction is
Derive an explicit form for and show that it is strongly peaked when for a reciprocal lattice vector.
State the Born approximation for when the scattering is due to a potential . Calculate the Born approximation for the case
Electrons with de Broglie wavelength are incident on a target composed of many randomly oriented small crystals. They are found to be scattered strongly through an angle of . What is the likely distance between planes of atoms in the crystal responsible for the scattering?
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Paper 1, Section II, I
2010 comment(a) Define what it means to say that is an equilibrium distribution for a Markov chain on a countable state space with Q-matrix , and give an equation which is satisfied by any equilibrium distribution. Comment on the possible non-uniqueness of equilibrium distributions.
(b) State a theorem on convergence to an equilibrium distribution for a continuoustime Markov chain.
A continuous-time Markov chain has three states and the Qmatrix is of the form
where the rates are not all zero.
[Note that some of the may be zero, and those cases may need special treatment.]
(c) Find the equilibrium distributions of the Markov chain in question. Specify the cases of uniqueness and non-uniqueness.
(d) Find the limit of the transition matrix when .
(e) Describe the jump chain and its equilibrium distributions. If is the jump probability matrix, find the limit of as .
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Paper 2, Section II, I
2010 comment(a) Let be the sum of independent exponential random variables of rate . Compute the moment generating function of . Show that, as , functions converge to a limit. Describe the random variable for which the limiting function coincides with .
(b) Define the queue with infinite capacity (sometimes written ). Introduce the embedded discrete-time Markov chain and write down the recursive relation between and .
Consider, for each fixed and for , an queue with arrival rate and with service times distributed as . Assume that the queue is empty at time 0 . Let be the earliest time at which a customer departs leaving the queue empty. Let be the first arrival time and the length of the busy period.
(c) Prove that the moment generating functions and are related by the equation
(d) Prove that the moment generating functions and are related by the equation
(e) Assume that, for all ,
for some random variables and . Calculate and . What service time distribution do these values correspond to?
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Paper 3, Section II, I
2010 commentCars looking for a parking space are directed to one of three unlimited parking lots A, B and C. First, immediately after the entrance, the road forks: one direction is to lot A, the other to B and C. Shortly afterwards, the latter forks again, between B and C. See the diagram below.

The policeman at the first road fork directs an entering car with probability to A and with probability to the second fork. The policeman at the second fork sends the passing cars to or alternately: cars approaching the second fork go to and cars to .
Assuming that the total arrival process of cars is Poisson of rate , consider the processes and , where is the number of cars directed to lot by time , for . The times for a car to travel from the first to the second fork, or from a fork to the parking lot, are all negligible.
(a) Characterise each of the processes and , by specifying if it is (i) Poisson, (ii) renewal or (iii) delayed renewal. Correspondingly, specify the rate, the holding-time distribution and the distribution of the delay.
(b) In the case of a renewal process, determine the equilibrium delay distribution.
(c) Given , write down explicit expressions for the probability that the interval is free of points in the corresponding process, .
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Paper 4, Section II, I
2010 comment(a) Let be an irreducible continuous-time Markov chain on a finite or countable state space. What does it mean to say that the chain is (i) transient, (ii) recurrent, (iii) positive recurrent, (iv) null recurrent? What is the relation between equilibrium distributions and properties (iii) and (iv)?
A population of microorganisms develops in continuous time; the size of the population is a Markov chain with states Suppose . It is known that after a short time , the probability that increased by one is and (if ) the probability that the population was exterminated between times and and never revived by time is . Here and are given positive constants. All other changes in the value of have a combined probability .
(b) Write down the Q-matrix of Markov chain and determine if is irreducible. Show that is non-explosive. Determine the jump chain.
(c) Now assume that
Determine whether the chain is transient or recurrent, and in the latter case whether it is positive or null recurrent. Answer the same questions for the jump chain. Justify your answers.
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Paper 1, Section II, C
2010 commentFor let
Assume that the function is continuous on , and that
as , where and .
(a) Explain briefly why in this case straightforward partial integrations in general cannot be applied for determining the asymptotic behaviour of as .
(b) Derive with proof an asymptotic expansion for as .
(c) For the function
obtain, using the substitution , the first two terms in an asymptotic expansion as . What happens as ?
[Hint: The following formula may be useful
-
Paper 3, Section II, C
2010 commentConsider the ordinary differential equation
subject to the boundary conditions . Write down the general form of the Liouville-Green solutions for this problem for and show that asymptotically the eigenvalues and , behave as for large .
-
Paper 4, Section II, C
2010 comment(a) Consider for the Laplace type integral
for some finite and smooth, real-valued functions . Assume that the function has a single minimum at with . Give an account of Laplace's method for finding the leading order asymptotic behaviour of as and briefly discuss the difference if instead or , i.e. when the minimum is attained at the boundary.
(b) Determine the leading order asymptotic behaviour of
as
(c) Determine also the leading order asymptotic behaviour when cos is replaced by in .
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Paper 1, Section I, D
2010 commentA system with coordinates , has the Lagrangian . Define the energy .
Consider a charged particle, of mass and charge , moving with velocity in the presence of a magnetic field . The usual vector equation of motion can be derived from the Lagrangian
where is the vector potential.
The particle moves in the presence of a field such that
referred to cylindrical polar coordinates . Obtain two constants of the motion, and write down the Lagrangian equations of motion obtained by variation of and .
Show that, if the particle is projected from the point with velocity , it will describe a circular orbit provided that .
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Paper 2, Section I, D
2010 commentGiven the form
for the kinetic energy and potential energy of a mechanical system, deduce Lagrange's equations of motion.
A light elastic string of length , fixed at both ends, has three particles, each of mass , attached at distances from one end. Gravity can be neglected. The particles vibrate with small oscillations transversely to the string, the tension in the string providing the restoring force. Take the displacements of the particles, , to be the generalized coordinates. Take units such that and show that
Find the normal-mode frequencies for this system.
-
Paper 3, Section I, D
2010 commentEuler's equations for the angular velocity of a rigid body, viewed in the body frame, are
and cyclic permutations, where the principal moments of inertia are assumed to obey .
Write down two quadratic first integrals of the motion.
There is a family of solutions , unique up to time-translations , which obey the boundary conditions as and as , for a given positive constant . Show that, for such a solution, one has
where is the angular momentum and is the kinetic energy.
By eliminating and in favour of , or otherwise, show that, in this case, the second Euler equation reduces to
where and . Find the general solution .
[You are not expected to calculate or
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Paper 4, Section I, D
2010 commentA system with one degree of freedom has Lagrangian . Define the canonical momentum and the energy . Show that is constant along any classical path.
Consider a classical path with the boundary-value data
Define the action of the path. Show that the total derivative along the classical path obeys
Using Lagrange's equations, or otherwise, deduce that
where is the final momentum.
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Paper 2, Section II, D
2010 commentAn axially-symmetric top of mass is free to rotate about a fixed point on its axis. The principal moments of inertia about are , and the centre of gravity is at a distance from . Define Euler angles and which specify the orientation of the top, where is the inclination of to the upward vertical. Show that there are three conserved quantities for the motion, and give their physical meaning.
Initially, the top is spinning with angular velocity about , with vertically above , before being disturbed slightly. Show that, in the subsequent motion, will remain close to zero provided , but that if , then will attain a maximum value given by
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Paper 4, Section II, D
2010 commentA system is described by the Hamiltonian . Define the Poisson bracket of two functions , and show from Hamilton's equations that
Consider the Hamiltonian
and define
where . Evaluate and , and show that and . Show further that, when is regarded as a function of the independent complex variables and of , one has
Deduce that both and are constant during the motion.
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Paper 1, Section I, H
2010 commentExplain what is meant by saying that a binary code is a decodable code with words of length for . Prove the MacMillan inequality which states that, for such a code,
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Paper 2, Section I,
2010 commentDescribe the standard Hamming code of length 7 , proving that it corrects a single error. Find its weight enumeration polynomial.
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Paper 3, Section I,
2010 commentWhat is a linear code? What is a parity check matrix for a linear code? What is the minimum distance for a linear code
If and are linear codes having a certain relation (which you should specify), define the bar product . Show that
If has parity check matrix and has parity check matrix , find a parity check matrix for .
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Paper 4, Section I, H
2010 commentWhat is the discrete logarithm problem?
Describe the Diffie-Hellman key exchange system for two people. What is the connection with the discrete logarithm problem? Why might one use this scheme rather than just a public key system or a classical (pre-1960) coding system?
Extend the Diffie-Hellman system to people using transmitted numbers.
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Paper 1, Section II, H
2010 commentState and prove Shannon's theorem for the capacity of a noisy memoryless binary symmetric channel, defining the terms you use.
[You may make use of any form of Stirling's formula and any standard theorems from probability, provided that you state them exactly.]
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Paper 2, Section II, H
2010 commentThe Van der Monde matrix is the matrix with th entry . Find an expression for as a product. Explain why this expression holds if we work modulo a prime.
Show that modulo if , and that there exist such that . By using Wilson's theorem, or otherwise, find the possible values of modulo .
The Dark Lord Y'Trinti has acquired the services of the dwarf Trigon who can engrave pairs of very large integers on very small rings. The Dark Lord wishes Trigon to engrave rings in such a way that anyone who acquires of the rings and knows the Prime Perilous can deduce the Integer of Power, but owning rings will give no information whatsoever. The integers and are very large and . Advise the Dark Lord.
For reasons to be explained in the prequel, Trigon engraves an st ring with random integers. A band of heroes (who know the Prime Perilous and all the information contained in this question) set out to recover the rings. What, if anything, can they say, with very high probability, about the Integer of Power if they have rings (possibly including the fake)? What can they say if they have rings? What if they have rings?
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Paper 1, Section I, D
2010 commentWhat is meant by the expression 'Hubble time'?
For the scale factor of the universe and assuming and , where is the time now, obtain a formula for the size of the particle horizon of the universe.
Taking
show that is finite for certain values of . What might be the physically relevant values of ? Show that the age of the universe is less than the Hubble time for these values of .
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Paper 2, Section I, D
2010 commentThe number density for a photon gas in equilibrium is given by
where is the photon frequency. By letting , show that
where is a constant which need not be evaluated.
The photon entropy density is given by
where is a constant. By considering the entropy, explain why a photon gas cools as the universe expands.
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Paper 3, Section I, D
2010 commentConsider a homogenous and isotropic universe with mass density , pressure and scale factor . As the universe expands its energy changes according to the relation . Use this to derive the fluid equation
Use conservation of energy applied to a test particle at the boundary of a spherical fluid element to derive the Friedmann equation
where is a constant. State any assumption you have made. Briefly state the significance of .
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Paper 4, Section I, D
2010 commentThe linearised equation for the growth of density perturbations, , in an isotropic and homogenous universe is
where is the density of matter, the sound speed, , and is the comoving wavevector and is the scale factor of the universe.
What is the Jean's length? Discuss its significance for the growth of perturbations.
Consider a universe filled with pressure-free matter with . Compute the resulting equation for the growth of density perturbations. Show that your equation has growing and decaying modes and comment briefly on the significance of this fact.
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Paper 1, Section II, D
2010 commentA star has pressure and mass density , where is the distance from the centre of the star. These quantities are related by the pressure support equation
where and is the mass within radius . Use this to derive the virial theorem
where is the total gravitational potential energy and the average pressure.
The total kinetic energy of a spherically symmetric star is related to by
where is a constant. Use the virial theorem to determine the condition on for gravitational binding. By considering the relation between pressure and 'internal energy' for an ideal gas, determine for the cases of a) an ideal gas of non-relativistic particles, b) an ideal gas of ultra-relativistic particles.
Why does your result imply a maximum mass for any star? Briefly explain what is meant by the Chandrasekhar limit.
A white dwarf is in orbit with a companion star. It slowly accretes matter from the other star until its mass exceeds the Chandrasekhar limit. Briefly explain its subsequent evolution.
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Paper 3, Section II, D
2010 commentThe number density for particles in thermal equilibrium, neglecting quantum effects, is
where is the number of degrees of freedom for the particle with energy and is its chemical potential. Evaluate for a non-relativistic particle.
Thermal equilibrium between two species of non-relativistic particles is maintained by the reaction
where and are massless particles. Evaluate the ratio of number densities given that their respective masses are and and chemical potentials are and .
Explain how a reaction like the one above is relevant to the determination of the neutron to proton ratio in the early universe. Why does this ratio not fall rapidly to zero as the universe cools?
Explain briefly the process of primordial nucleosynthesis by which neutrons are converted into stable helium nuclei. Letting
be the fraction of the universe's helium, compute as a function of the ratio at the time of nucleosynthesis.
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Paper 1, Section II, H
2010 comment(i) State the definition of smooth manifold with boundary and define the notion of boundary. Show that the boundary is a manifold (without boundary) with .
(ii) Let and let denote Euclidean coordinates on . Show that the set
is a manifold with boundary and compute its dimension. You may appeal to standard results concerning regular values of smooth functions.
(iii) Determine if the following statements are true or false, giving reasons:
a. If and are manifolds, smooth and a submanifold of codimension such that is not transversal to , then is not a submanifold of codimension in .
b. If and are manifolds and is smooth, then the set of regular values of is open in .
c. If and are manifolds and is smooth then the set of critical points is of measure 0 in .
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Paper 2, Section II, H
2010 comment(i) State and prove the isoperimetric inequality for plane curves. You may appeal to Wirtinger's inequality as long as you state it precisely.
(ii) State Fenchel's theorem for curves in space.
(iii) Let be a closed regular plane curve bounding a region . Suppose , for , i.e. contains a rectangle of dimensions . Let denote the signed curvature of with respect to the inward pointing normal, where is parametrised anticlockwise. Show that there exists an such that .
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Paper 3, Section II, H
2010 comment(i) State and prove the Theorema Egregium.
(ii) Define the notions principal curvatures, principal directions and umbilical point.
(iii) Let be a connected compact regular surface (without boundary), and let be a dense subset of with the following property. For all , there exists an open neighbourhood of in such that for all , where denotes rotation by around the line through perpendicular to . Show that is in fact a sphere.
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Paper 4, Section II, H
2010 comment(i) Let be a regular surface. Define the notions exponential map, geodesic polar coordinates, geodesic circles.
(ii) State and prove Gauss' lemma.
(iii) Let be a regular surface. For fixed , and points in , let , denote the geodesic circles around , respectively, of radius . Show the following statement: for each , there exists an and a neighborhood containing such that for all , the sets and are smooth 1-dimensional manifolds which intersect transversally. What is the cardinality of ?
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Paper 1, Section I, D
2010 commentConsider the 2-dimensional flow
where and are non-negative, the parameters and are strictly positive and . Sketch the nullclines in the plane. Deduce that for (where is to be determined) there are three fixed points. Find them and determine their type.
Sketch the phase portrait for and identify, qualitatively on your sketch, the stable and unstable manifolds of the saddle point. What is the final outcome of this system?
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Paper 2, Section I, D
2010 commentConsider the 2-dimensional flow
where the parameter . Using Lyapunov's approach, discuss the stability of the fixed point and its domain of attraction. Relevant definitions or theorems that you use should be stated carefully, but proofs are not required.
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Paper 3, Section I, D
2010 commentLet . The sawtooth (Bernoulli shift) map is defined by
Describe the effect of using binary notation. Show that is continuous on except at . Show also that has -periodic points for all . Are they stable?
Explain why is chaotic, using Glendinning's definition.
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Paper 4, Section I, D
2010 commentConsider the 2-dimensional flow
Use the Poincaré-Bendixson theorem, which should be stated carefully, to obtain a domain in the -plane, within which there is at least one periodic orbit.
-
Paper 3, Section II, D
2010 commentDescribe informally the concepts of extended stable manifold theory. Illustrate your discussion by considering the 2-dimensional flow
where is a parameter with , in a neighbourhood of the origin. Determine the nature of the bifurcation.
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Paper 4, Section II, D
2010 commentLet and consider continuous maps . Give an informal outline description of the two different bifurcations of fixed points of that can occur.
Illustrate your discussion by considering in detail the logistic map
for .
Describe qualitatively what happens for .
[You may assume without proof that
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Paper 1, Section II, B
2010 commentThe vector potential is determined by a current density distribution in the gauge by
in units where .
Describe how to justify the result
A plane square loop of thin wire, edge lengths , has its centre at the origin and lies in the plane. For , no current is flowing in the loop, but at a constant current is turned on.
Find the vector potential at the point as a function of time due to a single edge of the loop.
What is the electric field due to the entire loop at as a function of time? Give a careful justification of your answer.
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Paper 3, Section II, B
2010 commentA particle of rest-mass , electric charge , is moving relativistically along the path where parametrises the path.
Write down an action for which the extremum determines the particle's equation of motion in an electromagnetic field given by the potential .
Use your action to derive the particle's equation of motion in a form where is the proper time.
Suppose that the electric and magnetic fields are given by
where and are constants and .
Find given that the particle starts at rest at the origin when .
Describe qualitatively the motion of the particle.
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Paper 4, Section II, B
2010 commentIn a superconductor the number density of charge carriers of charge is . Suppose that there is a time-independent magnetic field described by the three-vector potential
Derive an expression for the superconducting current.
Explain how your answer is gauge invariant.
Suppose that for there is a constant magnetic field in a vacuum and, for , there is a uniform superconductor. Derive the magnetic field for .
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Paper 1, Section II, A
2010 commentWrite down the Navier-Stokes equation for the velocity of an incompressible viscous fluid of density and kinematic viscosity . Cast the equation into dimensionless form. Define rectilinear flow, and explain why the spatial form of any steady rectilinear flow is independent of the Reynolds number.
(i) Such a fluid is contained between two infinitely long plates at . The lower plate is at rest while the upper plate moves at constant speed in the direction. There is an applied pressure gradient in the direction. Determine the flow field.
(ii) Now there is no applied pressure gradient, but baffles are attached to the lower plate at a distance from each other , lying between the plates so as to prevent any net volume flux in the direction. Assuming that far from the baffles the flow is essentially rectilinear, determine the flow field and the pressure gradient in the fluid.
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Paper 2, Section II, A
2010 commentWhat is lubrication theory? Explain the assumptions that go into the theory.
Viscous fluid with dynamic viscosity and density is contained between two flat plates, which approach each other at uniform speed . The first is fixed at . The second has its ends at , where . There is no flow in the direction, and all variation in may be neglected. There is no applied pressure gradient in the direction.
Assuming that is so small that lubrication theory applies, derive an expression for the horizontal volume flux at , in terms of the pressure gradient. Show that mass conservation implies that , so that . Derive another relation between and by setting the pressures at to be equal, and hence show that
Show that lubrication theory applies if .
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Paper 3, Section II, A
2010 commentThe equation for the vorticity in two-dimensional incompressible flow takes the form
where
and is the stream function.
Show that this equation has a time-dependent similarity solution of the form
if and satisfies the equation
and is the effective Reynolds number.
Show that this solution is appropriate for the problem of two-dimensional flow between the rigid planes , and determine the boundary conditions on in that case.
Verify that has exact solutions, satisfying the boundary conditions, of the form
when . Sketch this solution when is large, and discuss whether such solutions are likely to be realised in practice.
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Paper 4, Section II, A
2010 commentAn axisymmetric incompressible Stokes flow has the Stokes stream function in spherical polar coordinates . Give expressions for the components of the flow field in terms of . Show that the equation satisfied by is
Fluid is contained between the two spheres , with . The fluid velocity vanishes on the outer sphere, while on the inner sphere . It is assumed that Stokes flow applies.
(i) Show that the Stokes stream function,
is the general solution of proportional to and write down the conditions on that allow all the boundary conditions to be satisfied.
(ii) Now let , with as . Show that with .
(iii) Show that when is very large but finite, then the coefficients have the approximate form
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Paper 1, Section I, E
2010 commentLet the complex-valued function be analytic in the neighbourhood of the point and let be the real part of . Show that
Hence find the analytic function whose real part is
-
Paper 2, Section I, E
2010 commentDefine
Using the fact that
where denotes the Cauchy principal value, find two complex-valued functions and which satisfy the following conditions
-
and are analytic for and respectively, ;
-
;
-
.
-
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Paper 3, Section I, E
2010 commentLet and denote the gamma and the zeta functions respectively, namely
By employing a series expansion of , prove the following identity
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Paper 4, Section , E
2010 commentThe hypergeometric function can be expressed in the form
for appropriate restrictions on .
Express the following integral in terms of a combination of hypergeometric functions
[You may use without proof that ]
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Paper 1, Section II, E
2010 commentConsider the partial differential equation for ,
where is required to vanish rapidly for all as .
(i) Verify that the PDE can be written in the following form
(ii) Define , which is analytic for . Determine in terms of and also the functions defined by
(iii) Show that in the inverse transform expression for the integrals involving may be transformed to the contour
By considering where and , show that it is possible to obtain an equation which allows to be eliminated.
(iv) Obtain an integral expression for the solution of subject to the the initialboundary value conditions of given .
[You need to show that
by an appropriate closure of the contour which should be justified.]
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Paper 2, Section II, 14E
2010 commentLet
where is a closed anti-clockwise contour which consists of the unit circle joined to a loop around a branch cut along the negative axis between and 0 . Show that
where
and
Evaluate using Cauchy's theorem. Explain how this may be used to obtain an analytic continuation of valid for all .
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Paper 1, Section II, 18H
2010 commentLet be a finite field with elements and its algebraic closure.
(i) Give a non-zero polynomial in such that
(ii) Show that every irreducible polynomial of degree in can be factored in as for some . What is the splitting field and the Galois group of over ?
(iii) Let be a positive integer and be the -th cyclotomic polynomial. Recall that if is a field of characteristic prime to , then the set of all roots of in is precisely the set of all primitive -th roots of unity in . Using this fact, prove that if is a prime number not dividing , then divides in for some if and only if for some integer . Write down explicitly for three different values of larger than 2 , and give an example of and as above for each .
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Paper 2, Section II, H
2010 comment(1) Let . What is the degree of ? Justify your answer.
(2) Let be a splitting field of over . Determine the Galois group . Determine all the subextensions of , expressing each in the form or for some .
[Hint: If an automorphism of a field has order 2 , then for every the element is fixed by .]
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Paper 3, Section II, H
2010 commentLet be a field of characteristic 0 . It is known that soluble extensions of are contained in a succession of cyclotomic and Kummer extensions. We will refine this statement.
Let be a positive integer. The -th cyclotomic field over a field is denoted by . Let be a primitive -th root of unity in .
(i) Write in terms of radicals. Write and as a succession of Kummer extensions.
(ii) Let , and . Show that can be written as a succession of Kummer extensions, using the structure theorem of finite abelian groups (in other words, roots of unity can be written in terms of radicals). Show that every soluble extension of is contained in a succession of Kummer extensions.
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Paper 4, Section II, H
2010 commentLet be a field of characteristic , and assume that contains a primitive cubic root of unity . Let be an irreducible cubic polynomial, and let be its roots in the splitting field of over . Recall that the Lagrange resolvent of is defined as .
(i) List the possibilities for the group , and write out the set in each case.
(ii) Let . Explain why must be roots of a quadratic polynomial in . Compute this polynomial for , and deduce the criterion to identify through the element of .
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Paper 1, Section II, B
2010 commentConsider a spacetime with a metric and a corresponding connection . Write down the differential equation satisfied by a geodesic , where is an affine parameter.
Show how the requirement that
where denotes variation of the path, gives the geodesic equation and determines .
Show that the timelike geodesics for the 2 -manifold with line element
are given by
where and are constants.
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Paper 2, Section II, B
2010 commentA vector field which satisfies
is called a Killing vector field. Prove that is a Killing vector field if and only if
Prove also that if satisfies
then
for any Killing vector field .
In the two-dimensional space-time with coordinates and line element
verify that and are Killing vector fields. Show, by using with the tangent vector to a geodesic, that geodesics in this space-time are given by
where and are arbitrary real constants.
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Paper 4, Section II, B
2010 commentThe Schwarzschild line element is given by
where and is the Schwarzschild radius. Obtain the equation of geodesic motion of photons moving in the equatorial plane, , in the form
where is proper time, and and are constants whose physical significance should be indicated briefly.
Defining show that light rays are determined by
where and may be taken to be small. Show that, to zeroth order in , a light ray is a straight line passing at distance from the origin. Show that, to first order in , the light ray is deflected through an angle . Comment briefly on some observational evidence for the result.
-
Paper 1, Section I, F
2010 commentExplain what it means to say that is a crystallographic group of isometries of the Euclidean plane and that is its point group. Prove the crystallographic restriction: a rotation in such a point group must have order or 6 .
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Paper 2, Section I, F
2010 commentShow that a map is an isometry for the Euclidean metric on the plane if and only if there is a vector and an orthogonal linear map with
When is an isometry with , show that is either a reflection or a glide reflection.
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Paper 3, Section I, F
2010 commentLet be a "triangular" region in the unit disc bounded by three hyperbolic geodesics that do not meet in nor on its boundary. Let be inversion in and set
Let be the group generated by the Möbius transformations and . Describe briefly a fundamental set for the group acting on .
Prove that is a free group on the two generators and . Describe the quotient surface .
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Paper 4, Section I, F
2010 commentDefine loxodromic transformations and explain how to determine when a Möbius transformation
is loxodromic.
Show that any Möbius transformation that maps a disc onto itself cannot be loxodromic.
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Paper 1, Section II, F
2010 commentFor which circles does inversion in interchange 0 and ?
Let be a circle that lies entirely within the unit Let be inversion in this circle , let be inversion in the unit circle, and let be the Möbius transformation . Show that, if is a fixed point of , then
and this point is another fixed point of .
By applying a suitable isometry of the hyperbolic plane , or otherwise, show that is the set of points at a fixed hyperbolic distance from some point of .
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Paper 4, Section II, F
2010 commentExplain briefly how Möbius transformations of the Riemann sphere are extended to give isometries of the unit ball for the hyperbolic metric.
Which Möbius transformations have extensions that fix the origin in ?
For which Möbius transformations can we find a hyperbolic line in that maps onto itself? For which of these Möbius transformations is there only one such hyperbolic line?
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Paper 1, Section II, F
2010 comment(a) Define the Ramsey number . Show that for all integers the Ramsey number exists and that .
(b) For any graph , let denote the least positive integer such that in any red-blue colouring of the edges of the complete graph there must be a monochromatic copy of .
(i) How do we know that exists for every graph ?
(ii) Let be a positive integer. Show that, whenever the edge of are red-blue coloured, there must be a monochromatic copy of the complete bipartite graph .
(iii) Suppose is odd. By exhibiting a suitable colouring of , show that .
(iv) Suppose instead is even. What is Justify your answer.
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Paper 2, Section II, F
2010 commentLet be a bipartite graph with vertex classes and . What does it mean to say that contains a matching from to ?
State and prove Hall's Marriage Theorem, giving a necessary and sufficient condition for to contain a matching from to .
Now assume that does contain a matching (from to ). For a subset , denotes the set of vertices adjacent to some vertex in .
(i) Suppose for every with . Show that every edge of is contained in a matching.
(ii) Suppose that every edge of is contained in a matching and that is connected. Show that for every with .
(iii) For each , give an example of with such that every edge is contained in a matching but for some with .
(iv) Suppose that every edge of is contained in a matching. Must every pair of independent edges in be contained in a matching? Give a proof or counterexample as appropriate.
[No form of Menger's Theorem or of the Max-Flow-Min-Cut Theorem may be assumed without proof.]
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Paper 3, Section II, F
2010 commentLet be a graph of order . Show that must contain an independent set of vertices (where denotes the least integer .
[Hint: take a random ordering of the vertices of , and consider the set of those vertices which are adjacent to no earlier vertex in the ordering.]
Fix an integer with dividing , and suppose that .
(i) Deduce that must contain an independent set of vertices.
(ii) Must contain an independent set of vertices?
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Paper 4, Section II, F
2010 commentState Euler's formula relating the number of vertices, edges and faces in a drawing of a connected planar graph. Deduce that every planar graph has chromatic number at most
Show also that any triangle-free planar graph has chromatic number at most 4 .
Suppose is a planar graph which is minimal 5 -chromatic; that is to say, but if is a subgraph of with then . Prove that . Does this remain true if we drop the assumption that is planar? Justify your answer.
[The Four Colour Theorem may not be assumed.]
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Paper 1, Section II, E
2010 commentDefine a Poisson structure on an open set in terms of an anti-symmetric matrix , where . By considering the Poisson brackets of the coordinate functions show that
Now set and consider , where is the totally antisymmetric symbol on with . Find a non-constant function such that
Consider the Hamiltonian
where is a constant symmetric matrix and show that the Hamilton equations of motion with are of the form
where the constants should be determined in terms of .
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Paper 2, Section II, E
2010 commentConsider the Gelfand-Levitan-Marchenko (GLM) integral equation
with , where are positive constants and are constants. Consider separable solutions of the form
and reduce the GLM equation to a linear system
where the matrix and the vector should be determined.
How is related to solutions of the equation?
Set where are constants. Show that the corresponding one soliton solution of the equation is given by
[You may use any facts about the Inverse Scattering Transform without proof.]
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Paper 3, Section II, E
2010 commentConsider a vector field
on , where and are constants. Find the one-parameter group of transformations generated by this vector field.
Find the values of the constants such that generates a Lie point symmetry of the modified equation ( )
Show that the function given by satisfies the KdV equation and find a Lie point symmetry of corresponding to the Lie point symmetry of which you have determined from .
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Paper 1, Section II, H
2010 commenta) State and prove the Banach-Steinhaus Theorem.
[You may use the Baire Category Theorem without proving it.]
b) Let be a (complex) normed space and . Prove that if is a bounded set in for every linear functional then there exists such that for all
[You may use here the following consequence of the Hahn-Banach Theorem without proving it: for a given , there exists with and .]
c) Conclude that if two norms and on a (complex) vector space are not equivalent, there exists a linear functional which is continuous with respect to one of the two norms, and discontinuous with respect to the other.
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Paper 2, Section II, H
2010 commentFor a sequence with for all , let
and for all and .
a) Prove that is a Banach space.
b) Define
and
Show that is a closed subspace of . Show that .
[Hint: find an isometric isomorphism from to
c) Let
Is a closed subspace of If not, what is the closure of
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Paper 3, Section II, H
2010 commentState and prove the Stone-Weierstrass theorem for real-valued functions.
[You may use without proof the fact that the function can be uniformly approximated by polynomials on
-
Paper 4, Section II, H
2010 commentLet be a Banach space.
a) What does it mean for a bounded linear map to be compact?
b) Let be the Banach space of all bounded linear maps . Let be the subset of consisting of all compact operators. Show that is a closed subspace of . Show that, if and , then .
c) Let
and be defined by
Is compact? What is the spectrum of Explain your answers.
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Paper 1, Section II, G
2010 commentShow that for all .
An infinite cardinal is called regular if it cannot be written as a sum of fewer than cardinals each of which is smaller than . Prove that and are regular.
Is regular? Is regular? Justify your answers.
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Paper 2, Section II, G
2010 commentLet be a non-zero ordinal. Prove that there exists a greatest ordinal such that . Explain why there exists an ordinal with . Prove that is unique, and that .
A non-zero ordinal is called decomposable if it can be written as the sum of two smaller non-zero ordinals. Deduce that if is not a power of then is decomposable.
Conversely, prove that if is a power of then is not decomposable.
[Hint: consider the cases ( a successor) and ( a limit) separately.]
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Paper 3, Section II, G
2010 commentDefine the sets . What is meant by the rank of a set?
Explain briefly why, for every , there exists a set of rank .
Let be a transitive set of rank . Show that has an element of rank for every .
For which does there exist a finite set of rank ? For which does there exist a finite transitive set of rank ? Justify your answers.
[Standard properties of rank may be assumed.]
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Paper 4, Section II, G
2010 commentState and prove the Completeness Theorem for Propositional Logic.
[You do not need to give definitions of the various terms involved. You may assume that the set of primitive propositions is countable. You may also assume the Deduction Theorem.]
Explain briefly how your proof should be modified if the set of primitive propositions is allowed to be uncountable.
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Paper 1, Section I, A
2010 commentA delay model for a population consists of
where is discrete time, and . Investigate the linear stability about the positive steady state . Show that is a bifurcation value at which the steady state bifurcates to a periodic solution of period 6 .
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Paper 2, Section , A
2010 commentThe population of a certain species subjected to a specific kind of predation is modelled by the difference equation
Determine the equilibria and show that if it is possible for the population to be driven to extinction if it becomes less than a critical size which you should find. Explain your reasoning by means of a cobweb diagram.
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Paper 3, Section I, A
2010 commentA population of aerobic bacteria swims in a laterally-infinite layer of fluid occupying , and , with the top and bottom surfaces in contact with air. Assuming that there is no fluid motion and that all physical quantities depend only on , the oxygen concentration and bacterial concentration obey the coupled equations
Consider first the case in which there is no chemotaxis, so has the spatially-uniform value . Find the steady-state oxygen concentration consistent with the boundary conditions . Calculate the Fick's law flux of oxygen into the layer and justify your answer on physical grounds.
Now allowing chemotaxis and cellular diffusion, show that the equilibrium oxygen concentration satisfies
where is a suitable normalisation constant that need not be found.
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Paper 4, Section I, A
2010 commentA concentration obeys the differential equation
in the domain , with boundary conditions and initial condition , and where is a positive constant. Assume and . Linearising the dynamics around , and representing as a suitable Fourier expansion, show that the condition for the linear stability of can be expressed as the following condition on the domain length
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Paper 2, Section II, A
2010 commentThe radially symmetric spread of an insect population density in the plane is described by the equation
Suppose insects are released at at . We wish to find a similarity solution to in the form
Show first that the PDE reduces to an ODE for if obeys the equation
where is an arbitrary constant (that may be set to unity), and then obtain and such that and for . Determine in terms of and . Sketch the function at various times to indicate its qualitative behaviour.
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Paper 3, Section II, A
2010 commentConsider an epidemic model in which is the local population density of susceptibles and is the density of infectives
where , and are positive. If is a characteristic population value, show that the rescalings reduce this system to
where should be found.
Travelling wavefront solutions are of the form , where and is the wave speed, and we seek solutions with boundary conditions . Under the travelling-wave assumption reduce the rescaled PDEs to ODEs, and show by linearisation around the leading edge of the advancing front that the requirement that be non-negative leads to the condition and hence the wave speed relation
Using the two ODEs you have obtained, show that the surviving susceptible population fraction after the passage of the front satisfies
and sketch as a function of .
-
Paper 1, Section II, G
2010 commentSuppose that is a square-free positive integer, . Show that, if the class number of is prime to 3 , then has at most two solutions in integers. Assume the is even.
-
Paper 2, Section II, G
2010 commentCalculate the class group of the field .
-
Paper 4, Section II, G
2010 commentSuppose that is a zero of and that . Show that . Show that , the ring of integers in , is .
[You may quote any general theorem that you wish, provided that you state it clearly. Note that the discriminant of is .]
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Paper 1, Section I, G
2010 comment(i) Let be an integer . Define the addition and multiplication on the set of congruence classes modulo .
(ii) Let an integer have expansion to the base 10 given by . Prove that 11 divides if and only if is divisible by 11 .
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Paper 2, Section I, G
2010 commentLet be an odd prime number. If is an integer prime to , define .
(i) Prove that defines a homomorphism from to the group . What is the value of
(ii) If , prove that .
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Paper 3, Section I, G
2010 comment(i) Let and be positive integers, such that is not a perfect square. If , show that every solution of the equation
in positive integers comes from some convergent of the continued fraction of .
(ii) Find a solution in positive integers of
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Paper 4, Section I, G
2010 commentLet be a prime number, and put
Prove that has exact order modulo for all , and deduce that must be divisible by a prime with . By making a suitable choice of , prove that there are infinitely many primes with .
-
Paper 3, Section II, G
2010 commentState precisely the Miller-Rabin primality test.
(i) Let be a prime , and define
Prove that is a composite odd integer, and that is a pseudo-prime to the base 2 .
(ii) Let be an odd integer greater than 1 such that is a pseudo-prime to the base 2 . Prove that is always a strong pseudo-prime to the base 2 .
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Paper 4, Section II, G
2010 commentLet be the set of all positive definite binary quadratic forms with integer coefficients. Define the action of the group on , and prove that equivalent forms under this action have the same discriminant.
Find necessary and sufficient conditions for an odd positive integer , prime to 35 , to be properly represented by at least one of the two forms
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Paper 1, Section II, A
2010 comment(a) State the Householder-John theorem and explain its relation to the convergence analysis of splitting methods for solving a system of linear equations with a positive definite matrix .
(b) Describe the Jacobi method for solving a system , and deduce from the above theorem that if is a symmetric positive definite tridiagonal matrix,
then the Jacobi method converges.
[Hint: At the last step, you may find it useful to consider two vectors and .]
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Paper 2, Section II, A
2010 commentThe inverse discrete Fourier transform is given by the formula
Here, is the primitive root of unity of degree , and
(1) Show how to assemble in a small number of operations if we already know the Fourier transforms of the even and odd portions of :
(2) Describe the Fast Fourier Transform (FFT) method for evaluating and draw a relevant diagram for .
(3) Find the costs of the FFT for (only multiplications count).
(4) For , using the FFT technique, find for and
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Paper 3, Section II, A
2010 commentThe Poisson equation in the unit square on , is discretized with the five-point formula
where and are grid points.
Let be the exact solution, and let be the error of the five-point formula at the th grid point. Justifying each step, prove that
where is some constant independent of .
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Paper 4, Section II, A
2010 commentAn -stage explicit Runge-Kutta method of order , with constant step size , is applied to the differential equation .
(a) Prove that
where is a polynomial of degree .
(b) Prove that the order of any -stage explicit Runge-Kutta method satisfies the inequality and, for , write down an explicit expression for .
(c) Prove that no explicit Runge-Kutta method can be A-stable.
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Paper 2, Section II, J
2010 comment(a) Suppose that
Prove that conditional on , the distribution of is again multivariate normal, with mean and covariance .
(b) The -valued process evolves in discrete time according to the dynamics
where is a constant matrix, and are independent, with common distribution. The process is not observed directly; instead, all that is seen is the process defined as
where are independent of each other and of the , with common distribution.
If the observer has the prior distribution for , prove that at all later times the distribution of conditional on is again normally distributed, with mean and covariance which evolve as
where
(c) In the special case where both and are one-dimensional, and , , find the form of the updating recursion. Show in particular that
and that
Hence deduce that, with probability one,
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Paper 3, Section II, J
2010 commentConsider an infinite-horizon controlled Markov process having per-period costs , where is the state of the system, and is the control. Costs are discounted at rate , so that the objective to be minimized is
What is meant by a policy for this problem?
Let denote the dynamic programming operator
Further, let denote the value of the optimal control problem:
where the infimum is taken over all policies , and denotes expectation under policy . Show that the functions defined by
increase to a limit Prove that . Prove that
Suppose that . Prove that .
[You may assume that there is a function such that
though the result remains true without this simplifying assumption.]
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Paper 4, Section II, J
2010 commentDr Seuss' wealth at time evolves as
where is the rate of interest earned, is his intensity of working , and is his rate of consumption. His initial wealth is given, and his objective is to maximize
where , and is the (fixed) time his contract expires. The constants and satisfy the inequalities , and . At all times, must be non-negative, and his final wealth must be non-negative. Establish the following properties of the optimal solution :
(i) ;
(ii) , where ;
(iii) for some constants and .
Hence deduce that the optimal wealth is
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Paper 1, Section II, E
2010 comment(a) Solve by using the method of characteristics
where is continuous. What is the maximal domain in in which is a solution of the Cauchy problem?
(b) Prove that the function
is a weak solution of the Burgers equation
with initial data
(c) Let be a piecewise -function with a jump discontinuity along the curve
and let solve the Burgers equation on both sides of . Prove that is a weak solution of (1) if and only if
holds, where are the one-sided limits
[Hint: Multiply the equation by a test function , split the integral appropriately and integrate by parts. Consider how the unit normal vector along can be expressed in terms of .]
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Paper 2, Section II, E
2010 comment(a) State the Lax-Milgram lemma. Use it to prove that there exists a unique function in the space
where is a bounded domain in with smooth boundary and its outwards unit normal vector, which is the weak solution of the equations
for the Laplacian and .
[Hint: Use regularity of the solution of the Dirichlet problem for the Poisson equation.]
(b) Let be a bounded domain with smooth boundary. Let and denote
The following Poincaré-type inequality is known to hold
where only depends on . Use the Lax-Milgram lemma and this Poincaré-type inequality to prove that the Neumann problem
has a unique weak solution in the space
if and only if .
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Paper 3, Section II, 30E
2010 commentConsider the Schrödinger equation
for complex-valued solutions and where is the Laplacian.
(a) Derive, by using a Fourier transform and its inversion, the fundamental solution of the Schrödinger equation. Obtain the solution of the initial value problem
as a convolution.
(b) Consider the Wigner-transform of the solution of the Schrödinger equation
defined for . Derive an evolution equation for by using the Schrödinger equation. Write down the solution of this evolution equation for given initial data .
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Paper 4, Section II, 30E
2010 commenta) Solve the Dirichlet problem for the Laplace equation in a disc in
using polar coordinates and separation of variables, . Then use the ansatz for the radial function.
b) Solve the Dirichlet problem for the Laplace equation in a square in
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Paper 1, Section II, C
2010 commentTwo states , with angular momenta , are combined to form states with total angular momentum
Write down the state with in terms of the original angular momentum states. Briefly describe how the other combined angular momentum states may be found in terms of the original angular momentum states.
If , explain why the state with must be of the form
By considering , determine a relation between and , hence find .
If the system is in the state what is the probability, written in terms of , of measuring the combined total angular momentum to bero?
[Standard angular momentum states are joint eigenstates of and , obeying
Units in which have been used throughout.]
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Paper 2, Section II, C
2010 commentConsider a joint eigenstate of and . Write down a unitary operator for rotation of the state by an angle about an axis with direction , where is a unit vector. How would a state with zero orbital angular momentum transform under such a rotation?
What is the relation between the angular momentum operator and the Pauli matrices when ? Explicitly calculate , for an arbitrary real vector , in this case. What are the eigenvalues of the operator ? Show that the unitary rotation operator for can be expressed as
Starting with a state the component of angular momentum along a direction , making and angle with the -axis, is susequently measured to be . Immediately after this measurement the state is . Write down an eigenvalue equation for in terms of . Show that the probability for measuring an angular momentum of along the direction is, assuming is in the plane,
where is a unit vector in the -direction. Using show that the probability that is of the form
determining the integers and in the process.
[Assume . The Pauli matrices are
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Paper 3, Section II, C
2010 commentWhat are the commutation relations between the position operator and momentum operator ? Show that this is consistent with being hermitian.
The annihilation operator for a harmonic oscillator is
in units where the mass and frequency of the oscillator are 1 . Derive the relation . Write down an expression for the Hamiltonian
in terms of the operator .
Assume there exists a unique ground state of such that . Explain how the space of eigenstates , is formed, and deduce the energy eigenvalues for these states. Show that
finding and in terms of .
Calculate the energy eigenvalues of the Hamiltonian for two harmonic oscillators
What is the degeneracy of the energy level? Suppose that the two oscillators are then coupled by adding the extra term
to , where . Calculate the energies for the states of the unperturbed system with the three lowest energy eigenvalues to first order in using perturbation theory.
[You may assume standard perturbation theory results.]
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Paper 4, Section II, C
2010 commentThe Hamiltonian for a quantum system in the Schrödinger picture is
where is independent of time. Define the interaction picture corresponding to this Hamiltonian and derive a time evolution equation for interaction picture states.
Let and be orthonormal eigenstates of with eigenvalues and respectively. Assume for . Show that if the system is initially, at , in the state then the probability of measuring it to be the state after a time is
to order .
Suppose a system has a basis of just two orthonormal states and , with respect to which
where
Use to calculate the probability of a transition from state to state after a time to order .
Show that the time dependent Schrödinger equation has a solution
Calculate the transition probability exactly. Hence find the condition for the order approximation to be valid.
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Paper 1, Section II, J
2010 commentThe distribution of a random variable is obtained from the binomial distribution by conditioning on ; here is an unknown probability parameter and is known. Show that the distributions of form an exponential family and identify the natural sufficient statistic , natural parameter , and cumulant function . Using general properties of the cumulant function, compute the mean and variance of when . Write down an equation for the maximum likelihood estimate of and explain why, when , the distribution of is approximately normal for large .
Suppose we observe . It is suggested that, since the condition is then automatically satisfied, general principles of inference require that the inference to be drawn should be the same as if the distribution of had been and we had observed . Comment briefly on this suggestion.
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Paper 2, Section II, J
2010 commentDefine the Kolmogorov-Smirnov statistic for testing the null hypothesis that real random variables are independently and identically distributed with specified continuous, strictly increasing distribution function , and show that its null distribution does not depend on .
A composite hypothesis specifies that, when the unknown positive parameter takes value , the random variables arise independently from the uniform distribution . Letting , show that, under , the statistic is sufficient for . Show further that, given , the random variables are independent and have the distribution. How might you apply the Kolmogorov-Smirnov test to test the hypothesis ?
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Paper 3, Section II,
2010 commentDefine the normal and extensive form solutions of a Bayesian statistical decision problem involving parameter , random variable , and loss function . How are they related? Let be the Bayes loss of the optimal act when and no data can be observed. Express the Bayes risk of the optimal statistical decision rule in terms of and the joint distribution of .
The real parameter has distribution , having probability density function . Consider the problem of specifying a set such that the loss when is , where is the indicator function of , where , and where . Show that the "highest density" region supplies a Bayes act for this decision problem, and explain why .
For the case , find an expression for in terms of the standard normal distribution function .
Suppose now that , that and that . Show that .
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Paper 4, Section II,
2010 commentDefine completeness and bounded completeness of a statistic in a statistical experiment.
Random variables are generated as , where are independently standard normal , and the parameter takes values in . What is the joint distribution of when ? Write down its density function, and show that a minimal sufficient statistic for based on is .
[Hint: You may use that if is the identity matrix and is the matrix all of whose entries are 1 , then has determinant , and inverse with .]
What is Is complete for
Let . Show that is a positive constant which does not depend on , but that is not identically equal to . Is boundedly complete for ?
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Paper 1, Section II, I
2010 commentState Carathéodory's extension theorem. Define all terms used in the statement.
Let be the ring of finite unions of disjoint bounded intervals of the form
where and . Consider the set function defined on by
You may assume that is additive. Show that for any decreasing sequence in with empty intersection we have as .
Explain how this fact can be used in conjunction with Carathéodory's extension theorem to prove the existence of Lebesgue measure.
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Paper 2, Section II, I
2010 commentShow that any two probability measures which agree on a -system also agree on the -algebra generated by that -system.
State Fubini's theorem for non-negative measurable functions.
Let denote Lebesgue measure on . Fix . Set and . Consider the linear maps given by
Show that and that . You must justify any assertion you make concerning the values taken by .
Compute . Deduce that is invariant under rotations.
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Paper 3, Section II, I
2010 commentLet be a sequence of independent random variables with common density function
Fix and set
Show that for all the sequence of random variables converges in distribution and determine the limit.
[Hint: In the case it may be useful to prove that , for all
Show further that for all the sequence of random variables converges in distribution and determine the limit.
[You should state clearly any result about random variables from the course to which you appeal. You are not expected to evaluate explicitly the integral
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Paper 4, Section II, I
2010 commentLet be a sequence of independent normal random variables having mean 0 and variance 1 . Set and . Thus is the fractional part of . Show that converges to in distribution, as where is uniformly distributed on .
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Paper 1, Section II, F
2010 comment(i) Let be a normal subgroup of the finite group . Without giving detailed proofs, define the process of lifting characters from to . State also the orthogonality relations for .
(ii) Let be the following two permutations in ,
and let , a subgroup of . Prove that is a group of order 12 and list the conjugacy classes of . By identifying a normal subgroup of of index 4 and lifting irreducible characters, calculate all the linear characters of . Calculate the complete character table of . By considering 6 th roots of unity, find explicit matrix representations affording the non-linear characters of .
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Paper 2, Section II, F
2010 commentDefine the concepts of induction and restriction of characters. State and prove the Frobenius Reciprocity Theorem.
Let be a subgroup of and let . We write for the conjugacy class of in , and write for the centraliser of in . Suppose that breaks up into conjugacy classes of , with representatives .
Let be a character of . Writing for the induced character, prove that
(i) if no element of lies in , then ,
(ii) if some element of lies in , then
Let and let , where and dihedral group and write down its character table. Restrict each -conjugacy class to and calculate the -conjugacy classes contained in each restriction. Given a character of , express Ind in terms of , where runs through a set of conjugacy classes of . Use your calculation to find the values of all the irreducible characters of induced to .
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Paper 3, Section II, F
2010 commentShow that the degree of a complex irreducible character of a finite group is a factor of the order of the group.
State and prove Burnside's theorem. You should quote clearly any results you use.
Prove that for any group of odd order having precisely conjugacy classes, the integer is divisible by 16 .
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Paper 4, Section II, F
2010 commentDefine the circle group . Give a complete list of the irreducible representations of
Define the spin group , and explain briefly why it is homeomorphic to the unit 3-sphere in . Identify the conjugacy classes of and describe the classification of the irreducible representations of . Identify the characters afforded by the irreducible representations. You need not give detailed proofs but you should define all the terms you use.
Let act on the space of complex matrices by conjugation, where acts by
in which denotes the block diagonal matrix . Show that this gives a representation of and decompose it into irreducibles.
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Paper 1, Section II, G
2010 commentGiven a lattice , we may define the corresponding Weierstrass -function to be the unique even -periodic elliptic function with poles only on and for which as . For , we set
an elliptic function with periods . By considering the poles of , show that has valency at most 4 (i.e. is at most 4 to 1 on a period parallelogram).
If , show that has at least six distinct zeros. If , show that has at least four distinct zeros, at least one of which is a multiple zero. Deduce that the meromorphic function is identically zero.
If are distinct non-lattice points in a period parallelogram such that , what can be said about the points
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Paper 2, Section II, G
2010 commentGiven a complete analytic function on a domain , describe briefly how the space of germs construction yields a Riemann surface associated to together with a covering map (proofs not required).
In the case when is regular, explain briefly how, given a point , any closed curve in with initial and final points yields a permutation of the set .
Now consider the Riemann surface associated with the complete analytic function
on , with regular covering map . Which subgroup of the full symmetric group of is obtained in this way from all such closed curves (with initial and final points ?
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Paper 3, Section II, G
2010 commentShow that the analytic isomorphisms (i.e. conformal equivalences) of the Riemann sphere to itself are given by the non-constant Möbius transformations.
State the Riemann-Hurwitz formula for a non-constant analytic map between compact Riemann surfaces, carefully explaining the terms which occur.
Suppose now that is an analytic map of degree 2 ; show that there exist Möbius transformations and such that
is the map given by .
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Paper 1, Section I, J
2010 commentConsider a binomial generalised linear model for data modelled as realisations of independent and logit for some known constants , and unknown scalar parameter . Find the log-likelihood for , and the likelihood equation that must be solved to find the maximum likelihood estimator of . Compute the second derivative of the log-likelihood for , and explain the algorithm you would use to find .
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Paper 2, Section I, J
2010 commentSuppose you have a parametric model consisting of probability mass functions . Given a sample from , define the maximum likelihood estimator for and, assuming standard regularity conditions hold, state the asymptotic distribution of .
Compute the Fisher information of a single observation in the case where is the probability mass function of a Poisson random variable with parameter . If are independent and identically distributed random variables having a Poisson distribution with parameter , show that and are unbiased estimators for . Without calculating the variance of , show that there is no reason to prefer over .
[You may use the fact that the asymptotic variance of is a lower bound for the variance of any unbiased estimator.]
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Paper 3, Section I, J
2010 commentConsider the linear model , where is a random vector, , and where the nonrandom matrix is known and has full column rank . Derive the maximum likelihood estimator of . Without using Cochran's theorem, show carefully that is biased. Suggest another estimator for that is unbiased.
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Paper 4, Section I, J
2010 commentBelow is a simplified 1993 dataset of US cars. The columns list index, make, model, price (in , miles per gallon, number of passengers, length and width in inches, and weight (in pounds). The data are displayed in as follows (abbreviated):

It is reasonable to assume that prices for different makes of car are independent. We model the logarithm of the price as a linear combination of the other quantitative properties of the cars and an error term. Write down this model mathematically. How would you instruct to fit this model and assign it to a variable "fit"?
provides the following (slightly abbreviated) summary:

Briefly explain the information that is being provided in each column of the table. What are your conclusions and how would you try to improve the model?
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Paper 1, Section II, J
2010 commentConsider a generalised linear model with parameter partitioned as , where has components and has components, and consider testing against . Define carefully the deviance, and use it to construct a test for .
[You may use Wilks' theorem to justify this test, and you may also assume that the dispersion parameter is known.]
Now consider the generalised linear model with Poisson responses and the canonical link function with linear predictor given by , where for every . Derive the deviance for this model, and argue that it may be approximated by Pearson's statistic.
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Paper 4, Section II, J
2010 commentEvery day, Barney the darts player comes to our laboratory. We record his facial expression, which can be either "mad", "weird" or "relaxed", as well as how many units of beer he has drunk that day. Each day he tries a hundred times to hit the bull's-eye, and we write down how often he succeeds. The data look like this:
\begin{tabular}{rrrr} \multicolumn{1}{l}{} & & & \ Day & Beer & Expression & BullsEye \ 1 & 3 & Mad & 30 \ 2 & 3 & Mad & 32 \ & & & \ 60 & 2 & Mad & 37 \ 61 & 4 & Weird & 30 \ & & & \ 110 & 4 & Weird & 28 \ 111 & 2 & Relaxed & 35 \ & & & \ 150 & 3 & Relaxed & 31 \end{tabular}
Write down a reasonable model for , where and where is the number of times Barney has hit bull's-eye on the th day. Explain briefly why we may wish initially to include interactions between the variables. Write the code to fit your model.
The scientist of the above story fitted her own generalized linear model, and subsequently obtained the following summary (abbreviated):

Why are ExpressionMad and Beer:ExpressionMad not listed? Suppose on a particular day, Barney's facial expression is weird, and he drank three units of beer. Give the linear predictor in the scientist's model for this day.
Based on the summary, how could you improve your model? How could one fit this new model in (without modifying the data file)?
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Paper 2, Section II, C
2010 commentConsider a 3-dimensional gas of non-interacting particles in a box of size where the allowed momenta are . Assuming the particles have an energy , calculate the density of states as .
Treating the particles as classical explain why the partition function is
Obtain an expression for the total energy .
Why is By considering the dependence of the energies on the volume show that the pressure is given by
What are the results for the pressure for non-relativistic particles and also for relativistic particles when their mass can be neglected?
What is the thermal wavelength for non-relativistic particles? Why are the classical results correct if the thermal wavelength is much smaller than the mean particle separation?
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Paper 3, Section II, C
2010 comment(i) Given the following density of states for a particle in 3 dimensions
write down the partition function for a gas of such non-interacting particles, assuming they can be treated classically. From this expression, calculate the energy of the system and the heat capacities and . You may take it as given that .
[Hint: The formula may be useful.]
(ii) Using thermodynamic relations obtain the relation between heat capacities and compressibilities
where the isothermal and adiabatic compressibilities are given by
derivatives taken at constant temperature and entropy, respectively.
(iii) Find and for the ideal gas considered above.
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Paper 4, Section II, C
2010 comment(i) Let be the probability that a system is in a state labelled by with particles and energy . Define
has a maximum, consistent with a fixed mean total number of particles , mean total energy and , when . Let and show that
where may be identified with the temperature and with the chemical potential.
(ii) For two weakly coupled systems 1,2 then and , . Show that where, if is stationary under variations in and for fixed, we must have .
(iii) Define the grand partition function for the system in (i) and show that
(iv) For a system with single particle energy levels the possible states are labelled by , where and . Show that
Calculate . How is this related to a free fermion gas?
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Paper 1, Section II, I
2010 commentWhat is a Brownian motion? State the reflection principle for Brownian motion.
Let be a Brownian motion. Let . Prove
for all . Hence, show that the random variables and have the same distribution.
Find the density function of the random variable .
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Paper 2, Section II, I
2010 commentWhat is a martingale? What is a supermartingale? What is a stopping time?
Let be a martingale and a supermartingale with respect to a common filtration. If , show that for any bounded stopping time .
[If you use a general result about supermartingales, you must prove it.]
Consider a market with one stock with prices and constant interest rate . Explain why an investor's wealth satisfies
where is the number of shares of the stock held during the th period.
Given an initial wealth , an investor seeks to maximize where is a given utility function. Suppose the stock price is such that where is a sequence of independent and identically distributed random variables. Let be defined inductively by
with terminal condition for all .
Show that the process is a supermartingale for any trading strategy .
Suppose is a trading strategy such that the corresponding wealth process makes a martingale. Show that is optimal.
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Paper 3, Section II, I
2010 commentConsider a market with two assets, a riskless bond and a risky stock, both of whose initial (time-0) prices are . At time 1 , the price of the bond is a constant and the price of the stock is uniformly distributed on the interval where is a constant.
Describe the set of state price densities.
Consider a contingent claim whose payout at time 1 is given by . Use the fundamental theorem of asset pricing to show that, if there is no arbitrage, the initial price of the claim is larger than and smaller than .
Now consider an investor with initial wealth , and assume . The investor's goal is to maximize his expected utility of time-1 wealth , where . Show that the optimal number of shares of stock to hold is .
What would be the investor's marginal utility price of the contingent claim described above?
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Paper 4, Section II, I
2010 commentConsider a market with no riskless asset and risky stocks where the price of stock at time is denoted . We assume the vector is not random, and we let and . Assume is not singular.
Suppose an investor has initial wealth , which he invests in the stocks so that his wealth at time 1 is for some . He seeks to minimize the subject to his budget constraint and the condition that for a given constant .
Illustrate this investor's problem by drawing a diagram of the mean-variance efficient frontier. Write down the Lagrangian for the problem. Show that there are two vectors and (which do not depend on the constants and ) such that the investor's optimal portfolio is a linear combination of and .
Another investor with initial wealth seeks to maximize his expected utility of time 1 wealth, subject to his budget constraint. Assuming that is Gaussian and for a constant , show that the optimal portfolio in this case is also a linear combination of and .
[You may use the moment generating function of the Gaussian distribution without derivation.]
Continue to assume is Gaussian, but now assume that is increasing, concave, and twice differentiable, and that and are of exponential growth but not necessarily of the form . (Recall that a function is of exponential growth if for some constants positive constants .) Prove that the utility maximizing investor still holds a linear combination of and .
[You may use the Gaussian integration by parts formula
where is a vector of independent standard normal random variables, and is differentiable and of exponential growth. You may also interchange integration and differentiation without justification.]
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Paper 1, Section I, F
2010 commentLet be a non-empty complete metric space with no isolated points, an open dense subset of and a countable dense subset of .
(i) Stating clearly any standard theorem you use, prove that is a dense subset of .
(ii) If is only assumed to be uncountable and dense in , does it still follow that is dense in ? Justify your answer.
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Paper 2, Section I, F
2010 comment(a) State the Weierstrass approximation theorem concerning continuous real functions on the closed interval .
(b) Let be continuous.
(i) If for each , prove that is the zero function.
(ii) If we only assume that for each , prove that it still follows that is the zero function.
[If you use the Stone-Weierstrass theorem, you must prove it.]
(iii) If we only assume that for each , does it still follow that is the zero function? Justify your answer.
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Paper 3, Section I, F
2010 commentLet and suppose that is complex analytic on an open subset containing .
(i) Give an example, with justification, to show that there need not exist a sequence of complex polynomials converging to uniformly on .
(ii) Let be the positive real axis and . Prove that there exists a sequence of complex polynomials such that uniformly on each compact subset of .
(iii) Let be the sequence of polynomials in (ii). If this sequence converges uniformly on , show that , where .
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Paper 4, Section I,
2010 commentFind explicitly a polynomial of degree such that
for every polynomial of degree . Justify your answer.
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Paper 2, Section II, 11F
2010 commentLet
, and
Let .
(i) State the Brouwer fixed point theorem on the plane.
(ii) Show that the Brouwer fixed point theorem on the plane is equivalent to the nonexistence of a continuous map such that for each .
(iii) Let be continuous, and suppose that
for each . Using the Brouwer fixed point theorem or otherwise, prove that
[Hint: argue by contradiction.]
(iv) Let . Does there exist a continuous map such that for each ? Justify your answer.
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Paper 3, Section II, F
2010 comment(i) Let be a continuous map with . Define the winding number of about the origin.
(ii) For , let be continuous with . Make the following statement precise, and prove it: if can be continuously deformed into through a family of continuous curves missing the origin, then .
[You may use without proof the following fact: if are continuous with and if for each , then .]
(iii) Let be continuous with . If is not equal to a negative real number for each , prove that .
(iv) Let and . If is continuous, prove that for each non-zero integer , there is at least one point such that .
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Paper 1, Section II, A
2010 commentDerive the wave equation governing the velocity potential for linearized sound waves in a compressible inviscid fluid. How is the pressure disturbance related to the velocity potential?
A semi-infinite straight tube of uniform cross-section is aligned along the positive -axis with its end at . The tube is filled with fluid of density and sound speed in and with fluid of density and sound speed in . A piston at the end of the tube performs small oscillations such that its position is , with and . Show that the complex amplitude of the velocity potential in is
Calculate the time-averaged acoustic energy flux in . Comment briefly on the variation of this result with for the particular case and .
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Paper 2, Section II, 38A
2010 commentThe equation of motion for small displacements in a homogeneous, isotropic, elastic medium of density is
where and are the Lamé constants. Show that the dilatation and rotation each satisfy wave equations, and determine the corresponding wave speeds and .
Show also that a solution of the form satisfies
Deduce the dispersion relation and the direction of polarization relative to for plane harmonic -waves and plane harmonic -waves.
Now suppose the medium occupies the half-space and that the boundary is stress free. Show that it is possible to find a self-sustained combination of evanescent -waves and -waves (i.e. a Rayleigh wave), proportional to exp and propagating along the boundary, provided the wavespeed satisfies
[You are not required to show that this equation has a solution.]
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Paper 3, Section II, 38A
2010 commentConsider the equation
where is a positive constant. Find the dispersion relation for waves of frequency and wavenumber . Sketch graphs of the phase velocity and the group velocity .
A disturbance localized near at evolves into a dispersing wave packet. Will the wavelength and frequency of the waves passing a stationary observer located at a large positive value of increase or decrease for ? In which direction do the crests pass the observer?
Write down the solution with initial value
What can be said about if is real?
Use the method of stationary phase to obtain an approximation for for fixed and large . What can be said about the solution at for large ?
[You may assume that for .]
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Paper 4, Section II, A
2010 commentStarting from the equations for one-dimensional unsteady flow of an inviscid compressible fluid, show that it is possible to find Riemann invariants that are constant on characteristics given by
where is the velocity of the fluid and is the local speed of sound. Show that for the case of a perfect gas with adiabatic equation of state , where and are constants, and when .
Such a gas initially occupies the region to the right of a piston in an infinitely long tube. The gas is initially uniform and at rest with density . At the piston starts moving to the left at a constant speed . Assuming that the gas keeps up with the piston, find and in each of the three distinct regions that are defined by families of characteristics.
Now assume that the gas does not keep up with the piston. Show that the gas particle at when follows a trajectory given, for , by
Deduce that the velocity of any given particle tends to as .
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Paper 3, Section I, D
2010 commentDerive the Euler-Lagrange equation for the function which gives a stationary value of
where is a bounded domain in the plane, with fixed on the boundary .
Find the equation satisfied by the function which gives a stationary value of
with given on .
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Paper 2, Section II, D
2010 commentDescribe briefly the method of Lagrange multipliers for finding the stationary points of a function subject to a constraint .
A tent manufacturer wants to maximize the volume of a new design of tent, subject only to a constant weight (which is directly proportional to the amount of fabric used). The models considered have either equilateral-triangular or semi-circular vertical crosssection, with vertical planar ends in both cases and with floors of the same fabric. Which shape maximizes the volume for a given area of fabric?
[Hint: ]
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Paper 4, Section II, D
2010 commentA function with given values of and makes the integral
stationary with respect to small variations of which vanish at and . Show that satisfies the first integral of the Euler-Lagrange equation,
for some constant . You may state the Euler-Lagrange equation without proof.
It is desired to tow an iceberg across open ocean from a point on the Antarctic coast (longitude ) to a place in Australia (longitude ), to provide fresh water for irrigation. The iceberg will melt at a rate proportional to the difference between its temperature (the constant , measured in degrees Celsius and therefore negative) and the sea temperature , where is the colatitude (the usual spherical polar coordinate . Assume that the iceberg is towed at a constant speed along a path , where is the longitude. Given that the infinitesimal arc length on the unit sphere is , show that the total ice melted along the path from to is proportional to
Now suppose that in the relevant latitudes, the sea temperature may be approximated by . (Note that is negative in the relevant latitudes.) Show that any smooth path which minimizes the total ice melted must satisfy
and hence that
where and are constants.
[Hint: